12 



For two-dimensional, incompressible flows the 

 stream-function-vorticity formulation is most 

 widely used in numerical fluid dynamics. It is 

 also a possible choice for the present investiga- 

 tions. It consists of the vorticity-transport 

 equation 



FIGURE 4. Downstream development of u' -disturbance at 

 y/Ay = 3 for boundary-layer flow disturbed periodically 

 (small amplitude) at upstream boundary. 



as observed in laboratory experiments should be 

 possible . 



For example, realistic numerical simulations of 

 Tollmien-Schlichting waves {as observed in the 

 Schubauer and Skramstad experiments) can be per- 

 formed by using at the upstream boundary A-D per- 

 iodic disturbances as produced by a vibrating ribbon 

 in the physical experiments. If the location of 

 a-D is considered to be somewhat downstream of the 

 ribbon in the real experiments , eigenf unctions of 

 linear stability theory may be conveniently used 

 to disturb the flow in the numerical simulation. 

 It was shown that the disturbance flow somewhat 

 downstream of the ribbon is well described by 

 linear stability theory when amplitudes are small. 



The disadvantage of the second approach is that 

 the development of numerical methods to solve the 

 resulting mathematical problem is considerably more 

 difficult than in the first approach. Although in 

 a strict mathematical sense both problems represent 

 mixed initial-boundary-value problems , the main 

 difference between the two concepts is that the 

 first approach results in a predominantly initial 

 value problem, where the temporal evolution of an 

 initially disturbed flow field is calculated. 



The second concept leads to a predominantly 

 boundary-value problem where the spatial reaction 

 of the flow field (which is also time-dependent, 

 of course) to disturbances introduced on the left 

 boundary is to be calculated. In the latter case 

 difficulties arise from the necessity of finding 

 adequate downstream boundary conditions which 

 allow unhindered passage of the disturbance waves 

 propagating downstream, and properly implementing 

 them into the numerical method. Since the aim of 

 this research effort is directed toward realistic 

 simulations of transition phenomena, emphasis in 

 the development of finite-difference methods was 

 placed on methods that were applicable to solving 

 the mathematical problem resulting from the latter 

 approach. The remainder of the discussions in this 

 paper are therefore also based on this concept. 



3. FORMULATIONS OF NAVIER-STOKES EQUATIONS FOR 

 NUMERICAL METHODS 



The Navier-Stokes equations can be cast into various 

 forms to be used as basis for a finite-difference 

 method. Each formulation has its inherent advan- 

 tages and disadvantages. The decision in favour of 

 a particular formulation has to be governed by the 

 physical flow problem to be investigated and by the 

 difference scheme finally used. In most cases, and 

 also particularly for the present investigations, 

 such a decision is difficult to make beforehand. 

 Extensive preliminary numerical experiments are 

 necessary before a decision can be made in favour 

 of a particular formulation. 



3 (1) 8 oj 3 0) 1 , 



— - + u r— + V -r = -— Ato 



3 t 3 X 3 y Re 



(1) 



and a Poisson equation for the stream function 



Ai|) = (1) , (2) 



where A is the Laplace operator, lo is defined as 

 3 u 3 "v 



3 y 3 X 

 and the stream function as 



(3) 



34) 

 3y 



u 



3 ji 

 3 X 



(4) 



With this definition of the stream function the 

 continuity condition 



3 u 3 V 



3-ir + 3-7 = ° 



(5) 



is satisfied for the continuum equations, however, 

 not necessarily for the discretized equations. All 

 variables in Eqs. (1) to (5) are dimensionless; 

 they are related to their dimensional counterparts , 

 denoted by bars , as follows 



L ' ^ L 



li)L 



Re = 



U L 

 



u 



i, = 



V 



tu 







UqL ' 



where L is a characteristic length, _Uo a reference 

 velocity and Re a Reynolds number (v kinematic vis- 

 cosity) . Thus this formulation represents a system 

 of two partial differential equations, each of 

 second order, for the unknown variables u) and if 

 because u and v in Eq. (1) can be eliminated using 

 Eq. (4). 



A variation of this formulation is the so-called 

 conservative form for which the vorticity-transport 

 equation 



3ti] 3 (uoi) 

 3t "^ 3x 



3 (vm) 

 3y 



1_ 

 Re 



Ao) 



(6) 



is used instead of Eq. (1) . With this formulation 

 conservation of vorticity is guaranteed for the 

 continuiom equations. 



A second formulation of the governing equations 

 also consists of a vorticity-transport equation (1) 

 or (6) . However, instead of the Poisson equation, 

 (2) , for ijj, two Poisson equations for the velocity 

 components u and v are used 



Au 



3 M 

 3y 



(7) 



Av = - 



3 iii 

 3 X 



