Boundary Lyer Theory

CHAPTER XV
Non-steady boundary layers t
a. Gencrnl remarks nn the cnlculntiott of non-wteady boundary layers
The oxamplos of solut,ions of the boundary-layer equations which havc been
considarcti until now rcfcrrctl to stcady motion. 't'hcy arc by far the most important
cases cncountcrctl in pmctic:al applications. Ncvcrtl~elcss, in this chapter we propose
to consitlnr scvcrnl examples of motions which tlcpcnd on time, i. c. of non-steady
1)ountIary Iaynrs.
The most comnlon oxnntplcs of non-stcatly bountlary layers occur whcn the
motion is slnrted /rom rest or whcn it is periodic. When motion is started from rest
both the body antl the fluid havc zero velocities up to a certain instant of time.
The motion begins at that instant and we can consider either that the body is
tlraggcd through the fluid at rcst or that the hods is at rcst and that the external
fluid motion varies with time. In this lattcr cam an initially very thin boundary
layer is formed near the body, and the transition from the vclocity of the body to
that in the extcrni~l flow takes place across it. Immediatcly after the start of the
motion t.hc flow in the whole fluid space is irrotstional antl potential with the exception
of a very thin Inyor ncar the body. Tho thickness of the boundary layer increases
with time, and it is important to investigate at which instant soparation (reverse
flow) first, occurs as tho I)ountlary hycr cont,in~~cs to build up. One such example
was ;rlrcatly consitlorccl in Src:. V 4; it was the exact solut,ion of the Navier-Stokes
equations for the flow noar a wall which is accclcmted impi~lsively from rest and
rnovcs in a clircct,ior~ p:~rallcl t.o itself. Also, the start of the flow in a pipe (See. V 6)
I)clongs to thc sanlc category.
1'11rthrr cxnn~ltlrs of non-stjcady I~ourtctary layers occur when eithcr the body
performs a poriotlir motion in a fluid at, rrst, or whcn the body is at rest and the
fluid rxoc:ut.rs a pcrioclic motion. The motion of a fluid ncar a wall which oscillates
in its own plnnc: (Srr. V 7) :~ffords an example of this type of problcm.
1. Ilnundnry-lnycr cquntinns. The funtlan~cnt~al equations for non-steady boundary
layers have already Occn tlcducetl in Scc. Vlla. In t11e general case when the flow is
c:omprcssil)lc and iron-stmtly I)ut two-tlimcnsional, ,we must resort to the following
cq~~a.tions for the vclorit.y n.ncl t.rmpcmturc ficltls (cf. eqns. (12.50a to e)):
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1. I nlll illtlrl,l.rcl l.o I'roScssor I