BAIL 2006 Book of Abstracts - Institut für Numerische und ...

M.H. BUSCHMANN, M. GAD-EL-HAK: Turbulent Boundary Layers: Reality and
Myth
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into the Navier-Stokes equations and taking the limit Re → ∞, leads to Prandtl’s boundary layer
equations
The boundary conditions are
uu x + vu y = ! p x + u yy ;
y = 0 :
u = v = 0 and
0 = p y ;
y ! " :
The task is now to find physically appropriate solution for eqn. (3).
ux + v = 0 (3)
y
u ! U e x ( ) (4)
What goals do we have when solving eqn. (3)? One of the main objectives is to find self-similar
solutions. Boundary layers are self-similar when normalization can be found so that data of
different physical realizations (e.g., experiments in different wind tunnels, profiles at different
downstream positions within one experiment) can be collapsed within one single curve.
Examples are the mean velocity profiles of the Blasius’ laminar zero-pressure-gradient boundary
layer and the fully-developed turbulent flow in pipes.
y
u( x, y )
u( x, y )
x 1
3 x
x 2
Which physical problems do we face when solving eqn. (3)? We neither know if for a certain
type of wall-bounded flow a transformation as searched for in the first question exists in general,
nor do we know what the transformation parameters are. The physically appropriate
transformation is a non-dimensionalization that is much more than simply changing the
coordinates. The crucial issue is top choose the scaling based on the physics of the problem. At a
minimum, the scale basis has to satisfy two criteria. It should consist of characteristic
parameters and represent problem-intrinsic scales. The foundation of dimensional analysis is the
Π-theorem formulated by Buckingham. 2
f ( x 1 ,x 2 ,...xn ) 0
Here xi denote the n variables of the system having m dimensions and Δi are the non-dimensional
similarity parameters of the problem.
2 Buckingham, E., “On physically similar systems: illustrations of the use of dimensional equations,” Phys. Rev., 2.
Ser., vol. 4, pp. 1119–1126, 1914.
u( x, y )
Π-theorem
y
!
Find proper
scaling parameters
Δ and U
= ( 1 2 n m)
u( x, y)
U
F ! , ! ,... ! " = 0
M. H. Buschmann & M. Gad-el-Hak, Abstract for BAIL 2006
Speaker: BUSCHMANN, M.H. 33 BAIL 2006
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