Smith 



useful for calculating unsteady flows at successive instants of time and for the 

 frequently occurring application of flow about a two-dimensional lifting airfoil. 



Maximum Element Number and Computation Times 



The maximum number of elements that may be used to approximate a body 

 surface has been largely a matter of arbitrary decisions made during program- 

 ming and does not represent any true limit. In two-dimensional and axisym- 

 metric cases, the maximum number of elements N is 400. Such a large number 

 has rarely proved necessary in practice. As was stated previously, most cases 

 of interest can be handled satisfactorily with less than 275 elements. Recently, 

 the old machine language program for three-dimensional flows has been re 

 placed by a FORTRAN IV program (20X). With it, up to 1000 unknown values of 

 a can be used to approximate the body, i.e., if the body has no plane of sym- 

 metry it can be approximated by 1000 elements. Provision is made in the pro- 

 gram to account for planes of symmetry. Therefore, if there is one plane, the 

 body can in effect be approximated by 2000 elements. For two and three planes, 

 the effective element numbers become 4000 and 8000 respectively. Most ap- 

 plications have at least one symmetry plane. Since the entire surface — not 

 just a single curve as in two-dimensional and axisymmetric cases— must be 

 approximated by elements, the element limits are somewhat lower than is de- 

 sirable. Very satisfactory results are obtained for single bodies if the shape 

 is not too extreme, but for interference problems the number of available ele- 

 ments is marginal. It is desirable to double the number of elements; beyond 

 this, little need for anything greater can be envisioned. 



Computing times are somewhat variable and depend on the geometry of the 

 body as well as on the number of elements. As a rough approximation, the com- 

 puting time is divided evenly between the calculation of the induced velocities 

 Vjj and the solution of the linear equations in Eq. (7). In axisymmetric cases 

 the calculation of induced velocities requires a somewhat greater fraction of the 

 time, and in two-dimensional cases requires somewhat less. The above division 

 of computing time varies considerably with changes in computing equipment. 



For the IBM 7094 computer, the following rough estimates of total computa- 

 tion time are useful for 100-element cases: two-dimensional bodies 1.6 min- 

 utes; axisymmetric bodies in axisymmetric flow, 2.6 minutes; and axisymmetric 

 bodies at angle of attack (including the axisymmetric flow), 4 minutes. These 

 estimates assume that only surface velocities are required. If the flow at a 

 large number of points off the body in the flow field is required, computing times 

 are increased. Of course, three-dimensional cases take much longer. Typical 

 computing times for cases of 650 elements are 1.5 hours on the IBM 7094. For 

 certain applications it is possible to reduce drastically the element number. 

 Useful results have been obtained in as little as 15 minutes. 



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