13 



which can be derived from the definition of vortic- 

 ity, (3) , using the continuity equation, (5). This 

 system of partial differential equations for the 

 u),u,v formulation is of higher order than the u,ijj 

 system. The higher order allows less restrictive 

 boundary conditions which is advantageous in appli- 

 cations to transition simulations as discussed in 

 Section 4. 



A third form of the governing equations is the 

 so-called primitive variable formulation with the 

 two momentum equations 



3 u 

 3 t 



+ u 



3 V 

 3t 



-— Au, 



(8) 



= - ^ + ^ Av, 



(where p = p/pUg, with density p) and a Poisson 

 equation for the pressure 



one has to keep in mind that the respective quanti- 

 ties (such as vorticity in the (i),i|) or a),u,v formu- 

 lation or momentum for the u,v,p formulation) are 

 initially only conserved for the continuum equations. 

 The conservation property may be carried over to 

 the discretized equations only if certain differ- 

 ence approximations (in this case, central differ- 

 ences) are used. For the implementation of the 

 boundary conditions it is frequently very difficult 

 or sometimes impossible to employ such difference 

 approximations required to maintain the conserva- 

 tion properties for the discretized equations. 



For the present investigations, comparison cal- 

 culations during the early stage of the development 

 of the numerical method have shown that, for the 

 10,4' or a),u,v systems, almost equivalent accuracy 

 can be obtained with either formulation. Because 

 the conservative formulation leads to a somewhat 

 slower solution algorithm for the solution of the 

 difference equations, preference was given there- 

 fore to a non-conservative formulation. 



, _ 3u 3 V 

 ^P = 2 i^ 3T 



3 V 3 u 

 3 x 3 y 



(9) 



Vorticity Transport (u,i^ or a),u,v) versus Primitive 

 Variable (u,v,p) Formulation 



which is derived from Eq. (8) using the continuity 

 Eq. (5). 



There is also a conservative form of the primi- 

 tive variable formulation (conserving momentum) 



3 u ^ a(u^) a (uv )_ _ ^ + ]_ ^^ 

 3 t 3x 3 y 3x Re 



(10) 



3 V ^ 3(uv) _^ 3(v^) 

 3 t 3x 3y 



3P 1 



-~- + — Av 



3y Re 



and a Poisson equation in a now different form 



,p = . 1!1h!L _ 2 4^ - 4^ - I? + ^ AD, (11) 



3x2 



3x3y 3y2 3t Re 



with the so-called dilation term 



D = 



3u 

 3x 



3y 



(12) 



The absence of the dilation terms in a Poisson equa- 

 tion for the pressure may cause nonlinear numerical 

 instability, which can be avoided when such terms 

 are retained (Harlow and Welch, 1965) . 



Conservative versus Nonconservative Formulation for 

 Use in Transition Studies 



The evaluation of the relative merits of conserva- 

 tive formulations over non-conservative ones is a 

 widely investigated subject in numerical fluid 

 dynamics [Roache (1976), Fasel (1978)]. Neverthe- 

 less, satisfactory answers have not yet been found 

 except for compressible flows for which conserva- 

 tive formulations are obviously advantageous. One 

 argument in favour of conservative formulations is 

 that better accuracy can be obtained. However, for 

 incompressible flow problems there are several ex- 

 amples contradicting this claim. When evaluating 

 possible advantages of a conservative formulation 



In reviewing literature on numerical simulations 

 of viscous incompressible flows it is noticeable 

 that formulations involving a vorticity-transport 

 equation, rather than the primitive variable form- 

 ulation, are preferred. The unpopularity of the 

 u,v,p system is a result of nimierable unsuccessful 

 attempts in applying it to calculations of viscous 

 incompressible flows. Although a few successful 

 applications based on the u,v,p system are reported 

 in more recent literature, there are still serious 

 arguments against its use for stability and trans- 

 ition simulations. Difficulties result from prob- 

 lems associated with the use of a Poisson equation 

 for the pressure. This equation is often a source 

 of numerical instabilities, possibly due to difficul- 

 ties of properly implementing the botindary conditions 

 for pressure into the numerical scheme. Although 

 the numerical instabilities could be brought under 

 control, at least to a degree, (for example by intro- 

 ducing the dilation terms in Eq. 11) so that solu- 

 tions could be obtained for steady flow problems, 

 the inherent inclination of this formulation to 

 numerical instability still prohibits its use for 

 transition simulations. Frequently numerical 

 solutions based on this system are of a slightly 

 oscillatory nature (although amplitudes are extremely 

 small) and therefore interaction with oscillations 

 of the physically meaningful disturbances as oc- 

 curring in transition studies cannot be avoided. 



For these reasons finite-difference methods de- 

 vised for investigations of stability and transition 

 are based on the equations in vorticity transport 

 form, i.e. either on the w,ii system (Eqs. 1 and 2) 

 or the u,u,v system (Eqs. 1 and 7) . Nevertheless 

 current efforts are also directed toward develop- 

 ment of difference methods based on the equations 

 in primitive-variable formulation. Emphasis is 

 placed on extreme numerical stability in order to 

 make this method also applicable for stability and 

 transition studies. The continuing attraction of 

 the equations in primitive-variable form results 

 from the fact that, for the three-dimensional case, 

 fewer fields of variables have to be stored than 

 for a vorticity-transport formulation. For the 

 three-dimensional case, storage requirements are an 



