A restricted class of mathematically admissible, unsteady, three dimensional flows was identified which may constitute part of the structure observed in turbulent boundary layers. The development of the model and some general results are discussed. The resulting solution has characteristics which suggest how upwelling low speed flow can trigger a downward jetting of irrotational high speed fluid into the boundary layer

Ash, R. L.

English

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DEPARTMENT OF MECHANICAL ENGINEERING AND MECHANICS
SCHOOL OF ENGINEERING
OLD DOMINION UNIVERSITY
NORFOLK, VIRGINIA
(NASA-CR-164036) A CLASS OF UNSTEADY,
TAsEE-DIMENSIONAL FLOW STRUCTURES IN
'iUhUULENT BUUNjARY LAYLhS F lOdl hepoCt, 1
Sep. 1979 - 3' A,_ 1. 1960 (Oid Uo®iuion
Uhiv., Njrcoli., Y,i.)	 12 n SIC A6i/MF A01
A CLASS OF UNSTEADY, THREE-DIMENSIONAL
FLOW STRUCTU'Z'FS IN TURBULENT BOUNDARY
LAY ER:i
By
Robert L. Ash, Principal Investigator
Final Report
For the period September 1, 1979 - August 31, 1980
h81- 19414
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Prepared for the
National Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia
Under	 ^^ 93a31/1^
Research Grant NSG 1634,^`^^ 4	 ^91
J.N. Hefner, Technical Monitor	 ^h	 i ?'j
High Speed Aerodynamics Division  
	
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DEPART14ENT OF MECHANICAL ENGINEERING AND MECHANICS
SCHOOL OF ENGINEERING
OLD DOMINION UNIVERSITY
NORFOLK, VIRGINIA
A CLASS OF UNSTEADY, THREE-DIMENSIONAL
FLOW STRUCTURES IN TURBULENT BOUNDARY
LAYERS
By
Robert L. Ash, Principal Investigator
Final Report
For the period September 1, 1979 - August 31, 1980
Prepared for the
National Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia 23665
Under
Research Crant NSG 1634
J.N. Hefner, Technical Monitor
High Speed Aerodynamics Division
ON
r^O	 Submitted by the
Old Dominion University Research Foundation
P.O. Box 6369
Norfolk, Virginia 23508
!March 1981
A CLASS OF UNSTEADY, THREE-DIMENSIONAL FLOW STRUCTURES
IN TURBULENT BOUNDARY LAYERS
By
Robert L. Ashl
SUMMARY
A restricted class of mathematically admissible, unsteady, three-
dimensional flows has been identified which may constitute part of the
structure observed in turbulent boundary layers. This report presents
the development of the model and discusses some general results. The
resulting solution has characteristics which suggest how upwelling low-
speed flow can trigger a downward jetting of irrotational high-speed
fluid into the boundary layer.
SYMBOLS
el• ,e2,e3	 unit vectors
P	 pressure
U	 local velocity vector
U-	free-stream velocity
u,v,w	 velocity components
V	 complex velocity function
y	 reciprocal streamwise disturbance length
t	 ratio of disturbance to free-stream velocity
V	 kinematic viscosity
&	 complex variable
0	 density
'Professor, Department of Mechanical Engineering and Mechanics, Old Dominion
University, Norfolk, Virginia 23508.
INTRODUCTION
Experimental investigations continue to show that turbulent boundary-layer
flows are composed of coherent, unsteady flow structures distributed randomly
in space and time (see ref. 1, for example). The mechanisms responsible for
the evolution and destruction of those structures are not understood. Neither
is the process responsible for the production and distribution of Reynolds
stresses. In order to modify turbulent flows in a favorable or controlled
manner, the physics of the flow must be better understood.
Presently, experimental flow visualization studies have shown that large,
unsteady vorticular structures are a major part of the outer turbulent
boundary layer (refs. 1 and 2). The actual form and size ,,r a single, identi-
fiable flow structure is unknown because there is no method available to
separate one structure from the remnants of all the others in which it is
immersed. Furthermore, Hama (ref. 3) has shown how visual observations of
unsteady streaklines can lead to erroneous conclusions and to improver isolation
and interpretation of structural features. The present investigation has
attempted to identify mathematically allowable solutions to the unsteady-three-
dimens'onal Navier-Stokes equations which may constitute part of a turbulent
boundary layer. The ultimate objective of this work is to enable experimental
identification of turbulent flow structures and develop techniques for
modifying turbulent flows favorably.
Flow structures—identified loosely as vortices—are apparently embedded
in turbulent flows. A variety of structures may be present, and none of them
are presumed so large that they are required to satisfy inner and outer
boundary conditions simultaneously. Rather, structures near the wall are
presumed to satisfy wall boundary conditions while outer structures must
satisfy the irrotational outer conditions. The approach employed here was
to attempt to find solutions to the constant property Navier-Stokes equations
which could be subjected to either the inner or outer boundary conditions
(but not both). Thus far, models appropriate to outer flow conditions appear
to show more promise than the near wall structures. The results of the outer
structure studies follow.
C
GOVERNING EQUATIONS
The local velocity vector u, given by
uiuel +ve t +we 3 	 (1)
where el , e2 , and e3
 are unit vectors in the x-, y- and z- directions of
a Eulerian coordinate, respectively, which must satisfy the Navier-Stokes
equations:
Continuity:
v • u-0
	 (2)
and Conservation of Momentum:
au
at + (u	 u - p VP + v7 2u	 (3)
where
^_ 
e; 8T
+e2 y +e3 a	 (4)
and P and a are pressure and density, respectively.
Referring to figure 1, a coordinate system is chosen which has its
origin at the nominal turbulent -nonturbulent interface. The boundary is
assumed to be initially parallel, and if no interfacial motion were present,
the velocity distribution for positive values of y would simply be
U - Um el	
(5)
In reality, the turbulent-nonturbulent interface is characterized by
three -dimensional peaks and valleys which occur rather abruptly, then are
obliterated slowly as they convect downstream (see ref. 1). The mathematical
model which follows has attempted to couple the low -speed burst disturbances
from the wall region with outer eddy structures. It is assumed that the low-
speed fluid parcel ejected from the wall beneath a flat plate, turbulent
3
boundary layer is not able to penetrate the turbulent-inviscid interface.
Rather, this slow moving fluid is surrounded by fine structure turbulence;
as it arrives at the upper reaches of the boundary layer, the fine structure
shields the streaky fluid from the free stream. The large velocity gradients
which must be present act to sustain the fine structure. In addition, the
upward moving filament causes a small bulge to occur in the turbulent-non-
turbulent interface region.
Consideration of pressure changes resulting from a bulge suggests from
Bernoulli's equation that, if the velocity above the disturbance increases,
the pressure decreases, drawing the bulge further upward. This type of
disturbance would tend to pull the turbulent fluid rapidly upward into the
irrotational regime as it is swept downstream. However, there appears to
be no stopping mechanism to control bulge height. An alternate model would
be to assume the inviscid fluid maintains a constant volume flow by pro-
ducing an interfacial depression to compensate for the bulge. Under those
conditions, the bulge need not produce a local velocity increase and the
destabilizing pressure disturbance need not occur. Those disturbances have
been studied here.
Consider the class of velocity distributions given by
U W U. el + v (x,Y,z,t)
 
e 2 + w (x •Y, z , t ) e 3	 (6)
where Um is assumed constant. Then, the continuity equations reduce to
a +a= 
'o
Ty
which can be satisfied when v and w are functions of x, t and complex
variable E, where
r =y+i z	 (8)ti
Then, equation (7) becomes
av
a 
	 Z	 (9)
(7)
4
which can be integrated to yield
w(x.E,t) n i v(x,E,t) + g(x,t)
	 (10)
Here, we choose g(x,t) = 0, since no apparent benefits result from Anclusion
of a function of x and t alone.
Equation (10) can be utilized in equations (3) and (6) to develop ex-
pressions for the conservation of momentum in the three -coordinate directions:
x or el direction:
a p = 0	 (11)
ax
y or e2 direction:
,
av + U . av =_
1ap az`	 (12)
at	 m ax	 a ac	
ax2
and z or e 3 direction:
	
av + U av = -.L ap * azv 1	 (13)i 
at	 m ax	 o a&	 ax2 J
Obviously, equations ( 12) and (13) are the same. Furthermore, the re-
quirement that ax = 0 implies that p n p(E,t), which requires that pressure
fluctuations consistent with the above equations be "global" in the x-
direction. Those pressure fluctuations are not particularly interesting, and
it is therefore assumed that 
a	
0. Hence, the equations of motion are
satisfied by
2
av + U ax =	
a v
	
(14)
ax2
where the nonlinear terms in the momentum equations are cancelled through the
relationship:
w = iv(x,^,t)	 (15)
r
5
The linear partial differential equation (14) can be solved using clas-
sical techniques, and if V is defined by
V n v - iw	 (16)
the solution to equation ( 14) subject to initial conditions:
V(x,iz,o) = v(x,i.-,u) - iw(x,iz,o)	 (17)
can be satisfied by
V . 1	 •xp(-( •ar•ia(x-^ , ^  f' f v(n,ic,o) exp Ii(WL•eNjdGd^dedW	 (18)(:w)' ff	 .!-..1.
where the overbars on the spatial variables have been made dimensionless by
U.( )
(^) _ (19)
and dimensionless time is given by
U2t
t n 	 (20)
Presently, the initial velocity disturbances which are considered global
through equation (18) are not as interesting us spatially periodic. That is,
if the disturbance is finite with respect to the spanwise coordinate, the
velocity distribution is given by
v = f(x,t) exp	 ( iz-^'	 (21)
and
w n if(x , t) exp [L" (it-y)]
	 (2Z)
where
	
is an unspecified spanwise wavelength, and f ( x,t) is a solution to
-F . ,F F	 ('5;
t	 "x	 :x-
6
with
F s U
	 (24)
m
The solution to F can be gotten using Fourier transforms as
F(x,L)	 2n fm 
e-9^t e-i8(x-t ) f^ eiBE F(E,o) dEd g 	(25)
At this point, solutions to F(x,t) are being studied for various types
of initial disturbances in the x-direction. The results of that investi-
gation will require considerably more computational time and development of
additional computer graphics capabilities in order to verify the presence of
these structures in a turbulent boundary layer.
One disturbance which has produced interesting results is
^Z
F(x,o)	 r e
_y x-
	 (26)
There, the integration of equation (2S) is possible and
Uu
	 e1 + r exp [2 , (iz-y) ( a ) 1 - (x-t)- (4t + 1 /y`) 1) (e. - iea) (2')
m	 2L + 1)1/=
Under those initial conditions (a small area of fluid is given an initial
upward velocity distribution at the turbulent-nonturbulent interface), the
model predicts that the bulge is accompanied by a laterally shifted downward
jetting of high-speed irrotational fluid into the boundary layer. This
process may approximate the inrush phase near the wall, which triggers new
bursts.
CONCLUSIONS
A mathematical model has been developed which predicts deformation of
the interface between a turbulent boundary layer and the outer, irrotational
flow. Although solutions can be developed which describe the full x-y t.lane
initial velocity conditions, those solutions do not appear to yield results
Fa
which do-tribe physics within the turbulent boundary layer. On the other
hand, a variety of restricted solutions can be developed which represent
surface deformations resulting from initial velocity disturbances in finite
regions of the x-y plane. Those calculations have shown that bulges at
the interface can produce jetting of irrotational fluid toward the wall which
may describe some turbulent features. Investigation of the finite area
disturbances should be pursued to permit either isolation or elimination of
the proposed structures as important features in a turbulent boundary layer.
8
REFERENCES
1. Smith, C.; and Abbot, P.E., eds.: Workshop on Coherent Structure of
Turbulent Boundary Layers. AFOSR/Lehigh Univ., 1976.
2. Falco, R.E.: Coherent Motions in the Outer Region of Turbulent Boundary
Layers. Phys. Fluids, Vol. 20, 1977, pp. 5124-5132.
3. Hama, F.R.: Streaklines in a Perturbed Shear Flow. Phys. Fluids, Vol.
5, 1962, pp. 644-650.
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A class of unsteady, three-dimensional flow structures in turbulent boundary layers

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