1、粘性流體的粘性流體的邊界元算邊界元算法法高效偉高效偉 教授 東南大學 工程力學系 Boundary Element Method inViscous FluidsExisting computational methods inviscous fluid mechanics Finite Difference Method (FDM) :- simple to use - fast for computation- requires regular mesh (structure mesh) Finite Volume Method (FVM): Finite Element Method (

2、FEM): Boundary Element Method (BEM)- uses integral form of basic equations of fluid mechanics- uses unstructure mesh- flux and gradient are not accurate- convenient for mesh generation- difficult to determine penalty parameters in penalty formulation - easy to set up computational model- small distu

3、rbance potential problems- incompressible fluid flowsFeatures of current boundary-domain integral equation method Solving full Navies-Stokes equations Valid for incompressible and compressible fluids for incompressible fluids, pressure term can be eliminated from the system equations Easy to be deve

4、loped as meshless BEM formulations using the Radial Integration Method (RIM) and, consequently, all advantages of BEM can be remained Uses primitive variables in basic equations Velocity gradients can be accurately determined (the accuracy is as high as velocity itself) Automatically satisfies infin

5、ite boundary condition, so advantageous for solving aerodynamic problems0iixutGoverning equations in viscous fluid flowsiuibContinuity equation:Momentum equation: fluid density: velocity: body force per unit mass: stress tensor: shear stress tensor0)(,iiutijijjjiibxxuutuij: heat conductivity: temper

6、ature: energy: pressure whereEnergy equations:iiijijiiiubJuuxxTkxtEorijkTEJp/pEJEquation of state:RTp(gas),AAppn00) 1(water)Constitutive relationship based onStokes hypothesis: viscosity: internal energy: specific heatijkkijjiijxuxuxu32: the Kronecker delta : outward normal to boundary: tractionwher

7、e)(3131zzyyxxiipjijintijijijp,21iiuueEeTcevvcitjnStress-pressure relationship:Stress-velocity relationship:Traction-stress relationship:Energy relationship:ijWeighted residual equation for conservation of momentumWeighted residual equationChoose weight function ijuto satisfy: is the Dirac delta func

8、tion:where)(xy )()()()(xfydxyyfdbxudxuutuujkjkijkkjjijduuudbuduttupduduuudnuuudtuujkjikkkijjijjijjijjijkjkijkkjijjij3/,0)(3/,ijkjikkkijxyuuBoundary-domain integral equation for conservation of momentum)(),()()()(),()()(),()()()()(),()()()()()(),()()(),()()(),()(,ydtuyxuydybyyxuydypyxuydyuyuyyxuydyuy

9、uyynyxuydyuyxtydytyxuxujijjijjijkjkijkjkijjijjijiFundamental solutions for momentum integral equationjiijjiijijrrrrrryxu,7161,)1ln(7161),(kkijjiijjiijrnrrrnrnryxt,)3,(),( 381),(kjiijkjikkijkijrrrrrrryxu,7161),(,rruijij8,3,/ )(/,rxyyrriiiifor 2Dfor 3Dwhereiirrxyr,)2() 3(and1Velocity divergence integr

10、al equation)(43)()()(43)(),()()()(),()()()()(),()()()()()(),()()(),()()(),()(,xpxuxuxydtuyxuydybyyxuydyuyuyyxuydyuyuyynyxuydyuyxtydytyxuxxujjjiijjiijkjkiijkjkiijjiijjiijii Pressure integral equations for internal points0iiiixuxutBased on continuity equation:txuxuxuxydtuyxuydybyyxuydyuyuyyxuydyuyuyyn

11、yxuydyuyxtydytyxuxpiijjjjjjkjkjkjkjjjjjlnln34)()()(1)(),()()()(),()()()()(),()()()()()(),()()(),()()(),()(, Pressure equations for boundary pointsBased on continuity equation:txutpiinlnln342nuxunii0iiiixuxutand the first invariant of strain rate:IIjIjxNxNuLxuxu),(),(2211Pressure for boundary points

12、can be expressed asIt can be seen that in general pressure is not equal to normal traction Weighted residual equation for conservation of energyWeighted residual equation for energyChoose weight function Tto satisfy: is the Dirac delta function:where)(xy )()()()(xfydxyyfdbuTdxwTdxTkxTdtETiiiiiiijiji

13、JuuwwheredTTkdnTTkdnTkTdxTkxTii20)(2xyTBoundary-domain integral equation for conservation of energy)()(),()()(),()()()(),()()(),()()(),()(,ydyubtEyxTydywyxTydywynyxTydyqyxTydykTyxTxkTiiiiiinwherenyTkyq)()(huuuuwjjijiji2/with being the enthalpyTChpFundamental solutions for energy integral equationDfo

14、ryxrDforyxryxT3),(412),(1ln21),()2() 3(iirryxT,21),(iinnrryxT,21),(where112for 2D and(i.e.,for 3D),/ )(/,rxyyrriiiixyrandNumerical implementation of steady incompressible flows0, 0, 0tutxjiCondition for steady incompressible flows:Discretization of boundary and domain:jjuNujjtNtpNpkjkjuuNuuwhere is

15、shape functions and is nodal values of .Njuju43761528(-1,-1)(0,1)(0,-1)(1,0)(1,-1)(1,1)(-1,1)(-1,0)Algebraic matrix equations for steady incompressible flows2uDpCYXAbbbbFor boundary nodes:2uDpCYXAuIIIIIFor internal nodes:2uDYXAppppFor pressurewhereX: containing boundary unknown velocities and tracti

16、onsTnnnnnnnnnuuuuuuuuuuuuuuuuuuu)(,)(,)(,)(,)(,)(2332223121212131312212131112112112Numerical example: Couette flowxyp=0p=5051100.20.40.60.8100.30.60.91.21.5uxyCurrentExactVelocity profile on vertical lines 123456Number of IterationsNorm of Residual101010101010Driven flow in an unitary square cavity

17、xyux=1, uy=0ux=0uy=011(0,0)Re=100 00.20.40.60.81-0.4-0.200.20.40.60.81uxyCurrentGhia et al-0.3-0.2-0.100.10.200.250.50.751xuyCurrentGhia et al-70-20308000.20.40.60.81xTractionTraction distribution along top wall Frame 001 10 Aug 2003 BEMRESULTSVelocity vectors 0102030Number of IterationsNorm of Resi

18、dual1010101010Vortex center: (0.6153, 0.7354) by current method (0.6172, 0.7344) by Ghia et al in1982Three-dimensional curved pipe flow r = 1p=PR=9p=0b=10=1 and =1 =50YXZFrame 001 03 Jun 2003 BEMRESULTSxyzTop cross-section planexyVertical central plane ux=0uy=0uz=0Lower end surfaceUpper end surfaceD

19、iscretization of half: 672 boundary elements 2880 linear cells 719 boundary nodes 2784 internal nodes 3503 nodes (total)Boundary conditions: Upper end: Lower end: 50.,0zyxttt. 0zyxtttFrame 001 28 Jan 2004 BEMRESULTSFrame 001 03 Jun 2003 BEMRESULTS12345768910191814131516171222112120232425262729283031

20、velocity vector plot for different sections over vertical central planeSectionxzuxuz p1 10.000-10.2501.9593-0.65042.0644-1.62623 8.8179-9.84881.5734-0.38901.62089.10215 7.7094-9.62401.7870-0.52571.862716.03897 6.6745-9.32551.8425-0.73861.985122.12049 5.7133-8.95341.8423-0.96102.077827.2860114.8257-8

21、.50761.8029-1.19302.161931.5646134.0118-7.98821.7290-1.43252.245335.0271153.2715-7.39521.6234-1.67502.332637.7730172.6048-6.72851.4890-1.91512.425939.9218192.0118-5.98821.3305-2.14772.526541.6027211.4924-5.17431.1538-2.37122.637042.9407231.0466-4.28670.9659-2.58672.761144.0433250.6745-3.32550.7752-2.80862.913744.9647270.3760-2