Numerical Solutions 



Laminar Boundary Layer on a Flat Plate 

 in a Flow with Disturbances 



O. F. Vasiliev and I. V. Pushkareva 

 Institute of Hydrodynamics 

 Siberian Department of the U.S.S.R. Academy of Sciences 

 Novosibirsk, U. S.S.R. 



presented by 



O. F. Vasiliev 



The work is devoted to the theoretical analysis of the behavior of the two- 

 dimensional laminar boundary layer along a flat plate when the free- stream ap- 

 proach flow of an incompressible fluid has disturbances. The influence both of 

 periodic disturbances of two types and of random disturbances of the simplest 

 type are treated. 



At first the boundary -layer velocity distribution is studied when the outside 

 stream u(x,t) has periodic disturbances imposed on a constant velocity flow uq . 

 As mentioned, this problem is treated in two variants. In the first case, 



u(x, t) = Up 



1 + \ COS a)[ — - t 

 'O 



(the disturbances are carried by the mean flow). 

 In the second case, 



u(t) = Uq (1 + A. cos cot) 



(the X axis is directed along the plate, t is the time). 



The assumption of the relative smallness of the disturbance amplitude \. 

 permits one to construct the solution in the form of an expansion in power series 

 of the small parameter \ . The coefficients of the first three terms of this series 

 were found. Because of a special choice of the nondimensional variables, the 

 problem is reduced to the determination of universal functions. 



Next the boundary-layer velocity fluctuations were studied, assuming that, 

 upon the free flow with the constant velocity uq are superimposed stochastic 

 disturbances u', carried by the free stream with the approach velocity uq , 



u(x,t) = Un + u' (t), t = t - — . 



^ ^ ' Uq 



It is thereby assumed that the flow velocity fluctuations are represented by a 

 stationary random function of t and that the relative intensity of turbulence in 

 the free flow is small: 



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