Finite Element Method - Boundary layer - inviscid flow coupling _appe

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

A few references on the topic of boundary layer-inviscid flow coupling are given in Chapter 6 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 1 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 2 Solve the boundary layer using an integral approach over an independently generated surface mesh If the surface nodes do 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 3 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

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-e I .2 shows

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 6 (Le Balleur and

coworkers)

In Fig E.2, Cp is the coefficient of pressure; s the coordinate along the surface; S

the boundary layer thickness; 0 the momentum thickness; C ’ the skin friction

coefficient; H the velocity profile shape parameter; p the density; V , the trans-

piration 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 , K * = E 2x0 ’

where n is the normal direction from the wing surface

entrainment model

We have the following equations to be solved in the integral boundary layer lag-

Continuity

u, ds

Momentum

- = - - ( H + 2 - M 2 ) - -

Lag-entrainment

O S = F [ L ( ( C T ) & - A q 5 ) + ( ) 0 du,

(1 + 0.1M2)

19 du,

u, ds where F is a function of C, and C ’ and given as

0.8Cf

3 0.o2ce + c,’ + -

(0.01 + C,)

F =

In the above equations, H and HI are the velocity profile shape parameters defined as

H = j / r ( l - % ) d n , HI =- s - s*

0

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

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