15 



stream into an undisturbed flow field. Denoting 

 tlie undisturbed flow field with capital letters the 

 initial conditions for the m,i() system can be written 

 as 



u)(x,y,0) = fi(x,y) 

 l^i(j(x,y,0) = >J'(x,y) 

 and for the a),u,v system 



''a)(x,y,0) = nU,Y) 

 <^ u(x,y,0) = U(x,y) 

 v(x,y,0) = V(x,y) 



(14) 



(15) 



(19) are applicable for the calculation of both the 

 steady, undisturbed and the unsteady, disturbed flow. 

 At the upstream bo>indary A-D the disturbances are 

 introduced by superimposing onto the profiles of a 

 basic, undisturbed flow (denoted by subscript B; for 

 example, Blasius profiles or Poiseuille profiles 

 could be used for the cases considered in Figures 

 1 and 2) so-called perturbation functions which are 

 dependent on y and t only. Thus for the w,ip formu- 

 lation we have 



u){0,y,t) = ug(y) + P„,(y,t) , 



tp(0,y,t) = ijjg{y) + P^(Y't) 



(20) 



and for the to,u,v formulation 



The undisturbed flow field is obtained by solving 

 the Navier-Stokes equations for the steady flow. 

 Of course , for the flow between two parallel plates 

 the Poiseuille profiles already represent exact 

 solutions of the Navier-Stokes equations and can 

 therefore be used directly. For the boundary-layer 

 flow a solution has to be calculated numerically 

 by solving the Navier-Stokes equations without the 

 unsteady term 3 u/Q t in Eq. (1) . The argument could 

 be raised that in this case Blasius profiles could 

 be used instead. The differences between the 

 Blasius solution and a numerical Navier-Stokes sol- 

 ution are indeed very small. Nevertheless, for 

 investigations with very small disturbance ampli- 

 tudes, the differences can be of the same order of 

 magnitude as the disturbances themselves and there- 

 fore the transient character of the flow could 

 become considerably distorted. The boundary condi- 

 tions used for the calculation of the undisturbed, 

 basic flow are discussed subsequently in connection 

 with the conditions used for the calculation of the 

 unsteady, disturbed flow. 



Boundary Conditions 



At solid walls (non-permeable, no-slip), such as 

 boundary A-B of Figure 1 or A-B and C-D of Figure 

 2, the velocity components vanish 



u=0 



=0 



U=0 



V=0 



(16) 



The vorticity-transport formulations (the u,v,p 

 formulation will not be discussed further) require 

 special treatment for the vorticity calculation at 

 the walls. For the u,4) formulation vorticity can 

 be calculated from the relationship 



3^1)) 

 3y2 



(17) 



derived from Eq. (2) 

 either 



for the a),u,v formulation 



3 01 



3y2 



derived from Eq. (7b) or 



3u 



(18) 



(19) 



resulting from Eq. (3) can be used. Equations (17)- 



01(0, y,t) = u (y) + P,,,(y,t) 

 B 



< u(0,y,t) = UB(y) + Pu(y,t) 



^v(0,y,t) = VB{y) + P^-(y,t) 



(21) 



For the calculation of the steady, undisturbed 

 flow field the perturbation functions in Eqs. (20) 

 and (21) of course vanish. For simulations of 

 Tollmien-Schlichting waves, for example, the 

 perturbation functions are periodic in time where 

 amplitude distributions (or so-called perturbation 

 profiles) as obtained from linear stability theory 

 can be used. 



The freestream boiindary C-D (Figure 1) for the 

 boundary-layer flow is an artificial boundary and 

 requires special considerations as discussed in 

 Section 2. For both the calculation of the steady 

 flow and the unsteady, disturbed flow, vorticity is 

 assumed zero (u'=S=0). For boundary-layer type flows, 

 vorticity for both basic and disturbance flow (when 

 disturbances are introduced within the boundary lay- 

 er) decays rapidly away from the wall and is practi- 

 cally zero at a distance of two <5 (6 boundary layer 

 thickness) from the wall. 



For the calculation of the steady flow using the 

 a),u,v system suitable conditions for C-D are 



U 



Uf3(x) 



(22) 



where the freestream velocity Uf g (x) may be speci- 

 fied according to the downstream pressure variation 

 of the boundary layer flow. A condition for the v 

 component can be derived from the continuity equa- 

 tion, (5) , using Eq. (22) 



3 V 

 3y 



dU^ (x) 

 fs 



dx 



(23) 



For the a),ijj system a condition equivalent to Eq. (22) 

 can be used 



3 y 



= U 



fs 



(x) 



(24) 



The >|)',u',v' disturbances decay relatively slowly 

 in direction normal to the wall. For example, for 

 Tollmien-Schlichting waves the i()' or v' amplitude 

 at 66*, (for Re*=530, based on displacement thick- 

 ness &*) may still be close to 50% of the maximal 

 amplitude. Therefore Dirichlet conditions {u'=v'= 

 i|)'=0) could only be used if the freestream boundary 

 were very far, for example 506*, from the wall. 



