Calculation of Thick Boundary Layer and 

 Near Wake of Bodies of 

 Revolution by a Differential Method 



V. C. Patel and Y. T. 

 The University of Iowa 

 Iowa City, Iowa 



Lee 



ABSTRACT 



The differential equations of the thick axisyinmetric 

 turbulent boundary layer and wake are solved using 

 a finite-difference method. The equations include 

 longitudinal and transverse surface curvature terms 

 as well as the static-pressure variation across the 

 boundary layer and wake. Closure of the mean-flow 

 equations is affected by a rate equation for the 

 Reynolds stress deduced from the turbulent kinetic- 

 energy equation. The results of the method are 

 compared with the two sets of data obtained at the 

 Iowa Institute of Hydraulic Research from experi- 

 ments in the tail region of a modified spheroid 

 and low-drag body of revolution, and also with the 

 predictions of a simple integral approach proposed 

 earlier. It is shown that the differential approach 

 is superior, provided due account is taken of the 

 normal pressure variation and the direct influence 

 of the extra rates of strain, associated with the 

 longitudinal and transverse surface curvatures, on 

 the length scale of the turbulence. 



1 . INTRODUCTION 



In the absence of flow separation, the boundary 

 layer on a pointed-tailed body of revolution con- 

 tinues to grow in thickness up to the tail. Over 

 the rear quarter of the length of a typical body, 

 the boundary layer thickness becomes large enough 

 to invalidate the assumptions of conventional thin 

 boundary-layer theory. The measurements of Patel, 

 Nakayama, and Damian (1974) on a modified spheroid 

 as well as those of Patel and Lee (1977) on a low- 

 drag body indicate that the breakdown of thin bound- 

 ary layer approximations is manifested by several 

 concurrent flow features, namely (a) the boundary 

 layer thickness is no longer small compared with 

 the local transverse and longitudinal radii of sur- 

 face curvature, (b) the velocity component normal 

 to the wall is not small, (c) the pressure is not 



constant across the boundary layer, and (d) the 

 pressure distribution on the body surface does not 

 conform with that predicted by potential flow theory, 

 as a consequence of the interaction between the 

 thick boundary layer and the external inviscid flow. 

 These features have been recognized in the develop- 

 ment of the simple integral method of Patel (1974) 

 for the calculation of a thick axisymmetric bound- 

 ary layer, and later on, in the formulation of the 

 interaction scheme of Nakayama, Patel, and Landweber 

 ( 1976a, b) which attempted to couple the boundary 

 layer, the near wake and the external inviscid flow 

 by means of successive iterations. Although the 

 overall iteration scheme proved to be quite success- 

 ful, the treatment of the boundary layer using the 

 integral method, and particularly its extension to 

 calculate the near wake, required many assumptions 

 which remain untested. The purpose of the present 

 work was therefore to develop a more rational pro- 

 cedure in which the differential equations of the 

 thick boundary layer and the near wake are solved 

 by means of a numerical method, since it appeared 

 that such a procedure would provide not only a 

 more reliable vehicle for the extension of the 

 boundary layer solution into the wake , but also 

 yield the detailed information on the velocity 

 profiles required for the interaction calculations. 

 This paper describes the new differential method 

 and evaluates its performance relative to the inte- 

 gral method as well as the available experimental 

 information. 



2. DIFFERENTIAL EQUATIONS AND TURBULENCE MODEL 



In the (x,y,i}>) coordinate system shown in Figure 1, 

 X and y are distances measured along and normal to 

 the body surface, respectively, and (fi is the azi- 

 muthal angle. As shown by Patel (1973) and Nakayeima, 

 Patel, and Landweber (1976b) , the momentum equa- 

 tions of a thick axisymmetric turbulent boundary 

 layer may be written 



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