Finite Element Method - Boundary layer - inviscid flow coupling _appe This monograph presents in detail the novel "wave" approach to finite element modeling of transient processes in solids. Strong discontinuities of stress, deformation, and velocity wave fronts as well as a finite magnitude of wave propagation speed over elements are considered. These phenomena, such as explosions, shocks, and seismic waves, involve problems with a time scale near the wave propagation time. Software packages for 1D and 2D problems yield significantly better results than classical FEA, so some FORTRAN programs with the necessary comments are given in the appendix. The book is written for researchers, lecturers, and advanced students interested in problems of numerical modeling of non-stationary dynamic processes in deformable bodies and continua, and also for engineers and researchers involved designing machines and structures, in which shock, vibro-impact, and other unsteady dynamics and waves processes play a significant role.
Appendix E Boundary layer-inviscid flow coupling A few references on the topic of boundary layer-inviscid flow coupling are given in Chapter In this appendix we shall briefly explain a simple procedure of this flow coupling procedure To understand the process of coupling the Euler and integral boundary solutions we shall consider a typical flow pattern around a wing as shown in Fig E.l Both turbulent and laminar regimes are shown in this figure We summarize the procedure as follows Step Solve the Euler equations in the domain considered around the aerofoil Here any mesh can be used independently of the mesh used for the boundary layer solution The solution thus obtained will give a pressure distribution on the surface of the wing Step Solve the boundary layer using an integral approach over an independently generated surface mesh If the surface nodes not coincide with the Euler mesh, the pressure needs to be interpolated to couple the two solutions The laminar portion near the boundary (Fig E.l) is calculated by the ‘Thwaites compressible’ method and the turbulent region is predicted by the ‘lag-entrainment’ integral boundary layer model Step The Euler and integral solutions are coupled by transferring the outputs from one solution to the other As indicated in Fig E 1, direct and semi-inverse couplings Fig El Flow past an aerofoil Typical problem for boundary layer-inviscid flow coupling Appendix E 303 Fig E2 Coupling techniques: (a) direct; (b) semi-inverse can be used for different regions The semi-inverse coupling is introduced here mainly to stabilize the solution in the turbulent region close to separation f~ipul-eI 2shows the flow diagrams for the present boundary layer-inviscid coupling Further details on the Thwaites compressible method and semi-inverse coupling can be found in the references discussed in Sec 6.12, Chapter (Le Balleur and coworkers) 304 Appendix E In Fig E.2, Cp is the coefficient of pressure; s the coordinate along the surface; S the boundary layer thickness; the momentum thickness; C ’ the skin friction coefficient; H the velocity profile shape parameter; p the density; V , the transpiration velocity; K* is a factor developed from stability analysis; the subscript ZI marks the viscous boundary layer region; S* the displacement thickness; the superscript i indicates inviscid region and the superscript m indicates the current iteration Following are useful relations for some of the above quantities: S* H = -0 , S * = / ~ ( l - ~ ) d n , d K * =2x0 E’ /?= a (E.l) where n is the normal direction from the wing surface We have the following equations to be solved in the integral boundary layer lagentrainment model Continuity u, ds Momentum du, u, ds d0 Cf -= (H+2-M2)-ds Lag-entrainment OS=F[L((CT)&-Aq5)+ ds H +HI ( ) du, u~ ds 19 du, ds (1 u, EQ + 0.1M2) where F is a function of C, and C ’ and given as 0.o2ce F= 0.8Cf + c,’+ (0.01 + C,) In the above equations, H and HI are the velocity profile shape parameters defined as H=j/r(l-%)dn, HI =-s - s* C , is the entrainment coefficient; ut)the mean component of the streamwise velocity at the edge of the boundary layer; M the Mach number; C, the shear stress coefficient; X the scaling factor on the dissipation length; the subscripts EQ and EQ, denote respectively the equilibrium conditions and equilibrium conditions in the absence of secondary influences on the turbulence structure Appendix E 305 Once the above equations are solved, the transpiration velocity V , is calculated as shown in Fig E.2 and is added to the standard Euler boundary conditions on the wall and plays the role of a surface source The coupling continues until convergence In practice, in one coupling cycle, several Euler iterations are carried out for each boundary layer solution ... close to separation f~ipul-eI 2shows the flow diagrams for the present boundary layer- inviscid coupling Further details on the Thwaites compressible method and semi-inverse coupling can be found... boundary layer lagentrainment model Continuity u, ds Momentum du, u, ds d0 Cf -= (H+2-M2)-ds Lag-entrainment OS=F[L((CT)&-Aq5)+ ds H +HI ( ) du, u~ ds 19 du, ds (1 u, EQ + 0.1M2) where F is a function... parameters defined as H=j/r(l-%)dn, HI =-s - s* C , is the entrainment coefficient; ut)the mean component of the streamwise velocity at the edge of the boundary layer; M the Mach number; C, the