粘性流体的 边界元算 法

1. 粘性流体的边界元算法 Boundary Element Method in Viscous Fluids 高效伟教授 东南大学 工程力学系

2. Existing computational methods inviscous fluid mechanics • Finite Difference Method (FDM): • simple to use • fast for computation • requires regular mesh (structure mesh) • Finite Volume Method (FVM): • uses integral form of basic equations of fluid mechanics • uses unstructure mesh • flux and gradient are not accurate • Finite Element Method (FEM): • convenient for mesh generation • difficult to determine penalty parameters in penalty formulation • Boundary Element Method (BEM) • easy to set up computational model • small disturbance potential problems • incompressible fluid flows

3. Features of current boundary-domain integral equation method • Uses primitive variables in basic equations • Solving full Navies-Stokes equations • Velocity gradients can be accurately determined • (the accuracy is as high as velocity itself) • Valid for incompressible and compressible fluids • for incompressible fluids, pressure term can be eliminated • from the system equations • Easy to be developed as meshless BEM formulations using • the Radial Integration Method (RIM) and, consequently, all • advantages of BEM can be remained • Automatically satisfies infinite boundary condition, • so advantageous for solving aerodynamic problems

4. Governing equations in viscous fluid flows or Continuity equation: Momentum equation: Energy equations: (gas), (water) Equation of state: k where : heat conductivity : temperature : energy : : pressure : fluid density : velocity : body force per unit mass : stress tensor : shear stress tensor T E J p

5. Constitutive relationship based onStokes’ hypothesis Stress-pressure relationship: Stress-velocity relationship: Energy relationship: Traction-stress relationship: where : viscosity : internal energy : specific heat : the Kronecker delta : outward normal to boundary : traction e

6. Weighted residual equation for conservation of momentum Weighted residual equation Choose weight function to satisfy: where is the Dirac delta function:

7. Boundary-domain integral equation for conservation of momentum

8. Fundamental solutions for momentum integral equation for 2D for 3D where and

9. Velocity divergence integral equation

10. Pressure integral equations for internal points Based on continuity equation:

11. Pressure equations for boundary points Based on continuity equation: and the first invariant of strain rate: Pressure for boundary points can be expressed as It can be seen that in general pressure is not equal to normal traction

12. Weighted residual equation for conservation of energy Weighted residual equation for energy where Choose weight function to satisfy: where is the Dirac delta function:

13. Boundary-domain integral equation for conservation of energy where with being the enthalpy

14. Fundamental solutions for energy integral equation for 2D and for 3D) where (i.e., and

15. Numerical implementation of steady incompressible flows Condition for steady incompressible flows: Discretization of boundary and domain: where is shape functions and is nodal values of .

16. Algebraic matrix equations for steady incompressible flows For boundary nodes: For internal nodes: For pressure where {X}: containing boundary unknown velocities and tractions

17. Numerical example: Couette flow Velocity profile on vertical lines

18. Driven flow in an unitary square cavity Re=100 Traction distribution along top wall

19. Velocity vectors Vortex center: (0.6153, 0.7354) by current method (0.6172, 0.7344) by Ghia et al in1982

20. Three-dimensional curved pipe flow =50 =1and =1

21. Discretization of half: 672 boundary elements 2880 linear cells 719 boundary nodes 2784 internal nodes 3503 nodes (total) Boundary conditions: Upper end: Lower end:

22. velocity vector plot for different sections over vertical central plane

23. Results of each section over central plane

24. Contour plot of pressure over vertical central plane

25. Iteration history for pipe flow

26. Computational time in different stages (Minutes)

27. Conclusion • Presented formulations are general, applicable to steady, unsteady, • compressible and incompressible flows. • No velocity gradients appear in the system of equations. • Pressure can be eliminated from the system of equations. • Velocity gradient can be explicitly derived from the basic integral • equation and, therefore, has high computational accuracy.