14 



order of magnitude even more critical than for the 

 two-dimensional calculations. 



was developed such that it is applicable with only 

 minor modifications for either formulation. 



Use of Navier-Stokes Equations for the Disturbance 

 Flow 



For stability and transition simulations, the depen- 

 dent variables, which appear in the different form- 

 ulations of the Navier-Stokes equations discussed 

 previously, are those of the total flow, that is, 

 including both the basic and the disturbance flow. 

 There is an alternate approach, namely, to decompose 

 the total flow into the basic flow and a disturbance 

 flow such that 



u=U+u' , v=V+v' , p=P+p' , ij)='i'+ip " , a)=n+io',(13) 



where the prime indicates the variables of the dis- 

 turbance flow and the capital letters denote those 

 of the basic flow. Substituting relationships (13) 

 into various forms of the Navier-Stokes equations, 

 it is possible to rewrite the equations with the 

 disturbance variables as dependent variables. Sev- 

 eral terms involving only the basic flow can be 

 dropped, assuming the basic flow satisfies the 

 Navier-Stokes equations . 



The aspect of directly solving the equations for 

 the disturbance variables is an attractive one, 

 since it is the disturbance conditions that are of 

 interest when performing numerical stability and 

 transition studies. For this reason this approach 

 has probably been preferred in earlier attempts. 

 It also allows for detailed investigations of the 

 effects of the nonlinear (convective) terms because, 

 in a difference method based on this form, the 'lin- 

 earization' can be conveniently switched on or off. 



A careful evaluation of this form of equations , 

 however, reveals that it also has some major disad- 

 vantages. The equations in disturbance form contain 

 several additional terms (involving disturbance 

 terms with terms of the basic flow) which are not 

 present in a corresponding formulation for the total 

 flow. Thus, in finite-difference solutions addi- 

 tional numerical operations are required. A more 

 serious disadvantage is that, because of the 

 additional terms involving the basic flow, the 

 basic flow quantities have to be kept in fast- 

 access computer storage to be readily accessible 

 for the numerical operations in order to avoid ex- 

 cessive computation times. On the other hand, 

 using the equations for the total flow the basic 

 flow quantities are not directly involved in the 

 solution algorithm. In this case they are only 

 required for analysis and better respresentation 

 of the results (for example to determine the dis- 

 turbance quantities) . For this purpose they can 

 be stored in mass storage of lower speed accessi- 

 bility. 



The availability of sufficient fast-access stor- 

 age is, even with the latest computer generation, 

 still a critical limitation for such numerical 

 investigations of stability and transition. For 

 large scale simulations involving large numbers of 

 grid points, use of the disturbance formulation is 

 prohibitive. For this reason, for the present re- 

 search effort, use of the equations for the total 

 flow variables was generally preferred instead of 

 the disturbance formulation. However, the basic 

 solution algorithm of the definite-difference method 



4. BOUNDARY AND INITIAL CONDITIONS 



The selection of adequate boundary conditions and 

 the practical implementation into a finite- 

 difference scheme represents one of the major dif- 

 ficulties in the development of a finite-difference 

 model applicable for stability and transition stud- 

 ies. Difficulties arise from the necessity that 

 boimdary conditions, selected and implemented along 

 the artificial boundaries (see Section 2) for the 

 finite integration domain, have to enable solutions 

 that would be identical to solutions if the govern- 

 ing equations were solved in the infinite domain. 

 There is, of course, no way of checking this be- 

 cause solutions for the infinite domain are not 

 available. This indicates that, for selecting 

 boundary conditions, it is necessary to rely on 

 experience, intuition, and test calculations. 



For practical reasons the boundary conditions 

 at these artificial boundaries have to be such that 

 physically meaningful results can be obtained with 

 a relatively small integration domain. The number 

 of grid points, and therefore computer storage and 

 amoiont of numerical operations required for a nu- 

 merical solution, is directly dependent on the size 

 of the integration domain. Thus, only with a rela- 

 tively small domain may the computational costs of 

 numerical simulations be kept within acceptable 

 limits. This aspect is of particular importance 

 during the testing phases of the numerical methods. 



There are also other difficulties resulting from 

 the complicated nature of the governing equations. 

 For the nonlinear systems of governing equations in 

 the formulations of Section 3 it is not yet possible 

 to decide if a given problem consisting of the 

 governing equations and a set of boundary conditions 

 is well-posed in the sense of Hadamard (1952) . More- 

 over, it is not obvious whether Hadamard' s postulates 

 for a well-posed problem are adequate to include 

 physically meaningful solutions only. Additional 

 difficulties may arise because finite-difference 

 methods frequently require more boundary conditions 

 than would be needed for the original differential 

 formulation if exact solutions were possible 

 [Richtmyer and Morton (1967)]. From numerical ex- 

 perimentation with model equations simpler than the 

 full Navier-Stokes equations it is known that these 

 additional 'niomerical' boundary conditions are of- 

 ten a source of numerical instabilities possibly 

 caused by certain inconsistencies. Therefore, one 

 is confronted with the delicate task of selecting 

 and implementing the extra conditons (where it is 

 normally not known a priori which conditions are 

 the extra ones) in such a way that the numerical 

 stability of an otherwise stable method would not 

 be adversely affected. 



Initial Conditions 



When the simulation of space dependent transition 

 phenomena is of interest as in the present in- 

 vestigation the reaction of the flow to disturbances 

 introduced at the upstream boundary has to be cal- 

 culated. In this case one may assume an undisturbed 

 flow as initial condition at t=0 enabling the dis- 

 turbance waves introduced for t>0 to propagate down- 



