A NUMERICAL METHOD FOR THE CALCULATION OF UNSTEADY …
72
A NUMERICAL METHOD FOR THE CALCULATION OF UNSTEADY LIFTING POTENTIAL FLOW PROBLEMS by BONG-JIN IM, B.E. A THESIS IN MECHANICAL ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE IN MECHANICAL ENGINEERING Approved Accepted August, 198 2
Transcript of A NUMERICAL METHOD FOR THE CALCULATION OF UNSTEADY …
A NUMERICAL METHOD FOR THE CALCULATION OF
UNSTEADY LIFTING POTENTIAL
FLOW PROBLEMS
by
BONG-JIN IM, B.E.
A THESIS
IN
MECHANICAL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE IN
MECHANICAL ENGINEERING
Approved
Accepted
August, 198 2
/I'C^ IP
^ '-^'^:&^'
//Of he-'
ACKNOWLEDGMENTS
I sincerely thank Dr. J. W. Oler for his direction of
this project and for much of my training in fluid dynamics.
I also express my appreciation to Dr. J. H. Strickland and
Dr. Raouf A. Ibrahim for their many helpful suggestions.
I further express my gratitude to the other graduate
students of Dr. Oler and Dr. Strickland's group, especially
to Tony G. Smith and Gary Graham, for without their unself
ish assistance on many aspects of this work, this project
would have been impossible.
11
TABLE OF CONTENTS
ACKNOWLEDGMENTS ii
TABLE OF CONTENTS iii
ABSTRACT iv
LIST OF FIGURES v
NOMENCLATURE vii
CHAPTER page
I. INTRODUCTION 1
1 . 1 Purpose and Scope of the Research 2
1.2 Review of Previous Research 2
II. FORMULATION OF THE POTENTIAL FLOW MODEL 5
2.1 The Mathematical Representation 5 2.2 Solution Method 8 2.3 Numerical solution by the Collocation
Method 11 2.4 Evaluation of the Influence Coefficients . 18 2.5 Calculation of Airloads 22 2.6 Separated Flow Model 24 2.7 Outline of Computer Program 34
III. RESULTS 37
3.1 Steady Flows 37 3.2 Unsteady Flows 45 3.3 Separated Flow 52
IV. CONCLUSIONS AND RECOMMENDATIONS 60
REFERENCES H
111
ABSTRACT
A potential flow model for two-dimensional airfoils in
unsteady motion with boundary layer separation is described.
The airfoil and wake surfaces are represented by a finite
set of uniform strength doublet panels. The doublet
strengths on the airfoil surface are determined by applying
a kinematic surface tangency condition to a Green's function
representation of the potential field, while simultaneously
enforcing the Kutta condition. Wake shedding is governed by
a dynamic free surface condition and the characteristics of
the flow near any boundary layer separation points. Wake
deformation is predicted by applying a geometric free sur
face condition.
Calculation results are presented for steady motion,
impulsively started rectilinear motion and sinusoidal pitch
oscillations .
IV
Figure
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
3.1
3.2
3.3
3.4
LIST OF FIGURES
Page
Finite Element Repersentation of Airfoil
and Wake Surfaces 13
The Kutta Condition at the Trailing Edge . . . . 14
Evaluation of Element Influence Coefficients . . 21
Separated Flow Model 25
The Bound Vorticity on an Airfoil 29
The Net Rate of Vorticity Shedding at the
Trailing Edge 31
The rate of Change of Potential Jump Across
the Trailing Edge 32
The Flow Chart 36
The Steady Pressure Distribution on a Circular
Cylinder 39
The Steady Pressure Distribution on a Joukowski
Airfoil 40
The Steady Pressure Distribution on a NACA0015
at 0 Angle of Attack 41
The Steady Pressure Distribution on a NACA0015
at 3 Angle of Attack il2
V
3.5
3.6
3.7
3.8
3.9
3.10
3. 11
3.12
3.13
3. 14
3.15
3. 16
The Steady Pressure Distribution on a NACA0015
at 6° Angle of Attack 43
The Steady Pressure Distribution on a NACA0015
at 11.3° Angle of Attack 44
Potential Jump Distribution on an Impulsively
Started Flat Plate 47
Indicial Lift and Circulation on an Impulsively
Started Flat Plate 48
The Trailing Wake Shape behind the Impulsively
Started Flat Plate 49
The Trailing Wake Shape behind the Periodically
Plunging Flat Plate 50
The Unsteady Pressure Distribution Development
on a NACA0015 at 0.1 rad. Angle of Attack . . . 51
Calculated Separated Wake Geometry(I) 53
Calculated Separated Wake Geometry(II) 54
The Calculation of The Airloads 56
Assumed Separation Point by Katz's Data . . . . 57
Velocity Profiles of Single Vortex 58
VI
NOMENCLATURE
A = Normal induced velocity coefficient
C = Potential influence coefficient
c = Chord length of the airfoil
CI = Lift coefficient
Cd = Drag coefficient
D = Normal downwash
P = Pressure
R = Distance between control point and field point,
r = Position vector of control point
S = Airfoil surface
TE = Trailing edge
t = time
U = Local velocity
Uco = Free stream velocity
W = Wake surface
D = Velocity potential
a = Doublet strength
= Unit normal vector on the airfoil surface
E, - Position vector of field point
CO = Angular velocity of the airfoil
vii
V
r = Circulation
Y = Vorticity
Vlll
CHAPTER I
INTRODUCTION
From airplane wings to automobiles, the prediction of
aerodynamic characteristics has been an important subject.
Aerodynamic optimization techniques have been developed for
military, aerospace, and commercial needs. Recently, the
effective utilization of wind energy has been the focus of
many aerodynamic studies.
In the design of airfoils, the maximum lift and stall
behaviour are among the most important characteristics.
They strongly influence the efficiency and maximum power
output of wind turbine power generation systems, and the
take-off and landing distances for airplanes.
Potential flow analysis has been the principle tool for
the prediction of aerodynamic characteristics of two-dimen
sional airfoils where viscous effects are limited to thin
boundary layers. In cases where the boundary layer becomes
thick or separates, classical potential airfoil theory must
be modified to account for the added vorticity in the flow.
The presently reported research has been directed at devel
oping a model capable of handling cases such as these.
J.J Purpose and Scope of the Research
The research described herein has been directed at de
veloping a mathematical model which can predict the aerody
namic loads for airfoils of arbitrary geometry in unsteady
and possibly, separated flows. The following is a list of
efforts which were conducted during the course of this work.
(a) Develop an unsteady aerodynamic airfoil model to be
used for both flat plates and airfoils with thick
ness in nonseparated flows.
(b) Adapt that mathematical model for airfoils at high
angles of attack with boundary layer separation.
(c) Verify the model with existing airfoil data.
1.2 Review of Previous Research
A historical review by Kraus(1978) of panel methods in
potential flow analysis reveals that one of the first uses
of this method was by A. M. S. Smith(1962) for an airfoil
with zero lift. Since that time, most applications of panel
methods have been for steady flows, although the general
method is suited to unsteady flows as is evidenced by the
works of AshleyC1966), Djojodihardjo and Widnall(1969),
Summa(1976), and 01er(1976). All of these investigations
have delt with nonseparating flows where the wake vorticity
is shed smoothly from a well defined trailing edge.
The potential flow analysis of separated flows has been
limited to bluff bodies for which conformal mapping
techniques may be applied. Sarpkaya's(1979) investigation
of the flow behind a circular cylinder is a typical example.
Finite element methods have not previously been applied to
bluff body problems involving boundary layer separation.
Katz(1979) has used a discrete vortex method to
represent the separated non-steady flow about a cambered
airfoil. His flow modeling is based on thin airfoil
theory. The separation point was assumed to be known from
experiment or from a separate boundary layer calculation.
The calculated results are in good agreement with the
corresponding experimental data.
The preceeding may be generalized by noting that panel
methods for nonseparated flows and discrete vortex methods
for separated bluff body flows have been advanced to the
point that acceptable engineering predictions may be made.
The present research is directed at the formulation of a
numerical model which incorporates both panel and discrete
vortex methods for the prediction of separated flows from
airfoil shapes.
CHAPTER II
FORMULATION OF THE POTENTIAL FLOW MODEL
2,J_ The Mathematical Representation
*
Consider the motion of a two-dimensional airfoil
through a homogeneous, incompressible, and inviscid fluid.
The airfoil surface is represented with respect to a sta
tionary coordinate system by,
S(r,t) = 0 . (2.1)
The wake following the airfoil may be defined by a surface
of potential discontinuity given by,
W(r,t) = 0 . (2.2)
The possibility of separated flow is accounted for by allow
ing the wake to include surfaces of potential discontinuity
emanating from a boundary layer separation point as well as
from the trailing edge.
Since the airfoil plus wake comprise a complete lifting
system and assuming that the ideal fluid was started from a
state of rest or uniform motion, it follows that the motion
is irrotational for all tim.es. This requirement of irrota-
tionality is a necessary and sufficient condition to guaran
tee the existence of a velocity potential, i.e..
V X u = 0 (2.3)
therefore,
u = u + ve (2.4)
Conservation of mass for an incompressible fluid re
quires that the vector velocity field not diverge. This re
quirement may be expressed as
U = 0. (2.5)
Substituing £q.(2.4) into Eq.(2.5) yields Laplace's equation
which is the governing equation for this flow, i.e..
V = 0. (2.6)
The solution to Eq.(2.6) may be obtained through appli
cation of the appropriate boundary conditions:
(1) The Infinity Condition
The disturbance potential resulting from the
presence of the airfoil must vanish at infinity.
(2) The Kinematic Surface Tangency Condition
On the airfoil surface, the normal relative fluid
velocity must be zero.
(3) The Kutta Condition
At all times, the flow of fluid from the trailing
edge must be smooth and continuous.
(4) The Boundary Layer Separation Condition
The sheet of potential discontinuity emanating from
a boundary layer separation point must reflect the
injection of the boundary layer vorticity.
(5) The Dynamic Free Surface Condition
The pressure must be continuous through the wake
surfaces, since they cannot sustain a load.
(6) The Geometric Free Surface Condition
The wake particles are convected downstream at the
local convection velocities.
Once the potential field has been determined, the
pressure distribution on the body may be found from
conservation of momentum which takes the familiar form of
Bernoulli's equation:
p = p^ . p | | A , 1 ,V„2} (2.")
8
2.2 Solution Method
As stated in the previous section, the governing equa
tion is the linear Laplace equation. By use of Green's
theorem, it may be shown (see Lamb, 1932, pp. 57 - 59) that
the velocity potential at any point in the flow may be given
by
o W ± dS (2.8)
a
W
= the potential doublet strength on S(^,t)=0
A(|)" r the potential doublet strength on W(I,t)=0 -r ->
V = the surface normal on S( i ,t)30 or W(Z,t)=0
R = the vector distance between the "field" point r,
and "source" point, i .
It should be noted that the infinity condition is inherently
satisfied by Eq.(2.8).
The kinematical surface tangency condition on the sur
face of the airfoil may be expressed (see Karamcheti, 1966,
pp. 190 - 192) as
1^ + li + 3 3 t 3 n
n = 0 (2.9)
on S(r,t) = ' n
->-n is a local surface normal and — represents the downwash
3t
on the airfoil due to the airfoil motion. It may be
expressed for a body fixed coordinate system as
3n at - (SR B ^ ^ n (2.10)
-> where U^ = airfoil translational velocity vector
CO = airfoil angular rotation vector. a Substituting into Eq.(2.9) yields
ad) / -> -^ ->•, n + U n = 0 (2.11)
which is valid for a body fixed reference frame.
Substituing Eq.(2.8) into Eq.(2.9) yields
^=rrr- f f a -—,-- — dS = 2n dndv \R /
n + . ^ /; A: 4-(i)ds! (2.12)
on S(r,t) = 0 .
This provides a governing integral equation for the unknown
doublet strength distributions on the airfoil and wake
surfaces. Once Eq.(2.12) has been solved subject to tr.e
remaining boundary conditions, the potential at any point ir,
the flow may be determined through Eq.(2.8).
10
The solution of equation Eq.(2.12) is made difficult by
the nonlinearity which arises from the fact that the doublet
distribution on the wake as well as the wake location is
dependent on the doublet distribution on the airfoil
surface. That is, the location of the wake at any instant
is a function of the previous velocity potential fields
which are also functions of the previous wake geometries.
A complete solution may be obtained by employing the
following step-by-step procedure.
(1) At t = 0, let the airfoil be started impulsively
and the freestream velocity brought instantaneously
to U^ with respect to the stationary coordinate
system. For this instant, there is no wake surface
and no contribution to the downwash on the airfoil
by the wake. A unique solution for the potential
field may be found through a simultaneous
application of the surface tangency (Eq. 2.9) and
Kutta conditions.
(2) Over the next infinitesimal time increment, assume
that the corresponding potential and velocity
fields are unchanged. As a result, the wake
surface generated during that time increment may be
predicted through application of the Kutta and
boundary layer separation conditions.
11
(3) For the next time step, the integral over the wake
surface in Eq.(2.12) is known and the equation may
once again be solved with the Kutta condition for
a .
(4) Again assuming the velocity field to remain
constant over the time increment, the new geometry
of the existing wake surface may be calculated
through application of the geometric free surface
condition. In addition, new wake elements are shed
as before.
(5) Steps (3) and (4) are repeated so that the solution
proceeds in a step-by-step manner towards the
steady state or periodic final result.
2.3 Numerical solution by the Collocation Method
In its exact integral form, Eq.(2.12) does not lend it
self to efficient solution by a digital computer. The situ
ation may be improved by applying a collocation or finite
element solution technique. For this purpose, the airfoil
and wake surfaces are discretized into M and N(t) elements,
respectively, as shown in Fig.2.1. Over each surface ele
ment, the unknown doublet strength distribution is
approximated with a uniform distribution of unknown
12
magnitude. In addition, the centroids of the surface
elements on the airfoil are idenfified as control points at
which the surface tangency condition is satisfied exactly.
In this way, the integral form of the surface tangency
condition given by Eq.(2.12), may be reformulated as M
simultaneous equations. Each equation represents the
application of the surface tangency condition to an
individual control point.
In addition to the surface tangency condition, a Kutta
condition must be applied at the trailing edge to uniquely
specify the net circulation about the airfoil. Although
there are many ways in which this condition may be applied,
the essential requirement of all forms of the Kutta
condition is that the flow proceeds smoothly from the
trailing edge of the airfoil. Actual enforcement of the
condition may be accomplished by specifying the direction of
wake shedding or by matching the upper and lower surface
trailing edge pressures (or velocities if a steady flow).
Whatever the method of application, the Kutta condition
provides an additional boundary condition which serves to
represent the essential consequence of viscous boundary
layers in a real fluid flow.
Consider the case of an isolated airfoil discretized
into M elements as shown in Fig.2.1. There are M surface
13
0 <¥• L.
-w—
CO
«•-0
c o
f-
+> a
V c 0 vi V L.
M 0) 0 (0
»•-i .
Q- 3 o m
Q:
+* c
0) ^ (0
O 2 £ 03 '
.—
u 0)
•»-»
•"• c
• w -
u.
•D c a
OJ
14
tc
CVJ I z
CVJ
D
u
c
Id
1} X
+> 10
c o
TO c o
(J
•p •p
3
X OJ
OJ
Lt.
15
tangency conditions plus the Kutta condition or M+1
equations. To satisfy the M+1 simultaneous relations, there
are only M unknown doublet panel strengths, a , so that the
problem, as stated, is overspecifled. Either an additional
singularity of unknown strength must be added to the flow or
the number of boundary conditions must be reduced. The
latter approach has been followed in the present
investigation .
Rather than applying surface tangency conditions on
both the upper and lower surface elements at the trailing
edge, the flow is required to be tangent to the trailing
edge bisector as shown in Fig.2.2. In this way, the Kutta
condition as well as approximate forms of the surface
tangency conditions at the trailing edge elements are
satisfied. Therefore, the three boundary conditions at the
trailing edge are replaced by a single one and the total
number of boundary conditions becomes M-1. A final
condition on the unknown doublet strengths is formed by
assuming that the potential jump across the trailing edge
has equal contributions from the upper and lower elements,
i.e. A4)_ = 0-0 and a - -a^ TE TE
/2.
With these approximations, the surface tangency
condition of Eq.(2.12) may be rewritten in matrix for-na as
[A(i,j)]ja.} = [Dj - [A^(I,D)] |A:''J1 (2.13)
16
where A(i,j) = normal induced velocity coefficient at
ith control point on airfoil surface due
to jth source element on airfoil surface.
1 ;2 2n ^J an.Bv. S. 1 J ^ ) -
A^(i,j) = normal induced velocity coefficient at
ith control point on airfoil surface due
to the jth source element on the wake. •
1 // 5'
3 -^
^IdW
a. = strength of the jth doublet element on
airfoil surface
W A(j). = strength of the jth doublet element on wake
Di = normal downwash at the ith control point due
to the freestream velocity and motion of the
airfoil
One may now recognize the product, [A(i,j)] ja.f , as
the normal induced velocity on S due to the disturbance
field created by the presence of S itself.
r W • 1 ( W)
[A (i,D)j]A4) { is the downwash on S due to the wake and
|D.| is the downwash due to the relative freestream fluid
and airfoil motions.
17
Recall that the airfoil and freestream are started
impulsively such that the wake doublet strengths, h^^'-| ,
are known for that instant and all later ones. Therefore,
it is convenient to define
Bi = total downwash array
so that Eq.(2.13) becomes
[A(i,j)] {c } = {B^} . (2.15)
This linear equation set may be solved for the airfoil
surface doublet distributions, {a.}, i.e..
\o.\ =[A(i,j)] -' \B.\ . (2.16)
It should be noted that the normal velocity influence
coefficient matrix, [A(i,j)], does not change with time
since the airfoil maintains a fixed geometry with respect to
a body fixed coordinate system.
Once the unknown doublet strengths have been
determined, the potential for any number of points in the
field may be found from a matrix expression of Eq.(2.8) or
U\ = [C(i,j)] ja } + [c"(i,j)] |.." I (2.1")
where C(i,j) = potential influence coefficient at the
ith control point due to the jth source
element on S
C^(i,j) = potential influence coefficient at the
ith control point due to the jth source
point on W
The potential influence coefficient matrix, [C(i,j)], like
the normal velocity influence coefficient matrix is
independent of time.
2.4 Evaluation of the Influence Coefficients
As noted in the previous section, the representation of
a general doublet distribution on a surface element by a
uniform distribution permits the definition of normal vel
ocity and potential influence coefficients. These were giv
en by
A(i,j) = normal induced velocity coefficient at the ith
control point of S due to the jth source element.
i_ / r _ f i ) dS 2n i' 5n3v \R
(2.17)
C(i,j) = potential influence coefficient at the ith con
trol point due to the jth source element.
2n g- .v \R i 1 dS (2.18)
19
The direct evaluation of the integrals of Eq.(2.17) and
(2.18) may be avoided by taking advantage of the analogy
between surface distributions of doublets and vortices. It
may be shown (Oler & Strickland, 1980, pp. 87 - 99) that a
general distribution of doublets may be represented by a
distribution of vortices on the surface. The strength of
the vortex sheet at any point is equal to the gradient of
the doublet strength with the vortices oriented normal to
that gradient. For the particular case of a surface element
having a uniform doublet distribution, an equivalent
representation is that of a vortex ring on the boundary of
the element with strength equal to the element doublet
strength. This is illustrated for a two-dimensional surface
element in Fig.2.3
The influence coefficients may be determined by
evaluating the vortex equivalents of the doublet elements.
Referring to Fig.2.3, the potential influence coefficient
may be written as
, . . , el 62 C(i,D ) ^ ^ - —.
2r tan
-1 / 1 y - tan . T • e - 1 / 2
(2.19)
ri-ex/ V^2*ex^J
The normal velocity influence coefficient, may written as
A(i,j) = •1 e.
2 71 2n r n
2 ' -•
(2.20)
2n
e xr z 1
r 1
e xr^ z 2 I -^ I n
Since the doublet strength for the elements in the wake
are known for all times, it is possible to utilize an
equivalent discrete vortex representation of the influence
due to the wake.
The discrete vortex approach is only a slight variation of
the doublet panel method and offers the benefit of a
significant reduction in computation time.
As described in the previous paragraphs, the net
influence of a two-dimensional uniform strength doublet
panel is equivalent to the combined influences of discrete
vortices at its boundaries with strengths equal to plus or
minus the panel strength. For two adjacent doublet panels,
the contribution to the total influence by the vortex at
their common boundary is proportional to the difference in
the two panel strengths. Since the strengths of the wake
elements are known for all times in the step-by-step
solution procedure, it is possible to replace the panelized
wake with an equivalent discrete vortex representation. An
21
%=-(r
Fig. 2.3 Evaluation of Element Influence Coefficients
22
important advantage of this approach is that the influence
of an individual vortex on a particular control point is
calculated only once instead of twice as would be required
in the finite element representation. This is particularly
important for the wake calculations since the wake geometry
changes with each time step and all wake influence
coefficients must be recalculated.
2.5 Calculation of Airloads
The pressure at any point in an irrotational, ideal
flow may be found with the unsteady Bernoulli equation.
P = P. - P J M - I ^^^^)^|- (2.21)
As shown by Summa(1976), this may be rewritten in body fixed
coordinates as
P(r,t) = P - P oo
3 o (r , t)
t -i-
[U -\J^'Z{t)-r^'l?{r,t) + i[vMr,t)]2}. (2.22)
The primary difference in Eq.(2.21) and Eq.(2.22) is the
additional increment to 3t on the body due to the ' .ction
of the body through the potential field. The significance
23
of this term may be appreciated by recognizing that even if
the potential field was steady, d(^/dt would not be equal to
zero unless the potential field were also uniform.
Eq.(2.22) may be rewritten in a more convenient form
for numerical computation by expressing v^ as
9 6-^ ^ s^ 8n n (2.23)
where v is the surface gradient. Recall that the surface s
tangency condition was written in body fixed coordinates as
a 4) Tn H = -{ CO X ?) n (2.2U)
With Eq.(2.22) , Eq.(2.23) may be expanded to yield
P = P - p CX)
H + (3„ - "B - B " ' • 's*
9n °° B B X r ) n
^ ('s'' ^1 3^ \9) (2.25)
By substituing Eq.(2.24) into Eq.(2.26), we arrive at
p = p (3* (V - u B
- 03 B
X r) V i> S^
24
- T [(U^ - U^ - .. X ? B B
) . n]
^ 1 (v^^)^} (2.26)
Eq.(2.26) provides the advantage of reducing the computation
of the gradient of the disturbance potential to the
computation of its surface gradient. With Eq.(2.26), the
airloads on the airfoils may be calculated by integrating
the pressure force vector components over the surfaces.
2.6 Separated Flow Model
For modeling purposes, it is assumed that the wake may
be adequately represented by two sheets of potential discon
tinuity. One surface extends from the trailing edge while
the other originates at the boundary layer separation point
as illustrated in Fig.2.4.
The rates at which vorticity is shed into the two wake
surfaces are related by the Kelvin-Helmholtz theorem to the
rate of change of the vorticity bound to the airfoil
surface. The theorem requires that the rate of change of
net vorticity in the flow field is zero, i.e.,
25
-J^-G
F i g . 2 . 4 S e p a r a t e d F l o u Model
or
26
dV net 3t
= 0
arb 3rw srs ^ bt 3t
(2.27)
Here, the net vorticity has been divided into three
components: the vorticity bound to the airfoil surface,
r, , the vorticity shed from the boundary layer separation
point, VQ , and the vorticity shed from the trailing edge,
r,, . The time derivatives of r,, and r represent the rate
of vorticity shedding to the respective wake surfaces.
A simple vorticity flux analysis may be utilized to
estimate the vorticity shedding rate from the boundary layer
separation point.
dr 6 / V i U at f^u[^ - ^ ] d y ^VI^ ^
5 1 |_ (u2) dy 2 9y (2.28)
U
2 •
Ue is the velocity at the edge of the boundary layer or the
surface velocity calculated by the potential flow routine.
27
The assumption is made that 100^ of the vorticity contained
in the boundary layer is injected into the inviscid flow
field at the separation point.
If "b is the vorticity per unit length along the
airfoil surface and a is the distribution of potential
discontinuity or doublet strength along the surface, then,
referring to Fig.2.5, the bound vorticity may be written as
^b = A^^ ^b ^S
= /^ dS A- dS °^ (2.29)
= Ac TE •
The rate of change of bound vorticity may be expressed in
terms of the difference in surface doublet strength at the
trailing edge or potential jump across the trailing edge:
b ^ £_ dt dt " TE •
(2.30)
Substituting Eq.(2.28) and (2.30) into Eq.(2.27)
provides an expression for the rate of shedding of vorticity
to the wake surface extending from the trailing edge.
dr w dt
d_ dt
( a_ ) + TE
U 2
2 I (2.3')
28
Recall that Eq.(2.31) was based on the Kelvin-Helmholtz
vorticity conservation theorem. The same result may be
arrived at by applying the dynamic free surface boundary
condition at the airfoil trailing edge. Specifically, the
pressure difference across the infinitely thin surface must
be zero since it cannot sustain a load. This leads to
P - P, = 0 u 1
It ^^ - h^ ^ 1 (*u - '^V ' (2.32)
Recognizing that 7cf) = u for the fluid fixed reference
frame, then
2 u u|) = 3t ^^"TE^ (2.33)
From Fig.2.6, it is noted that (u2-u2)/2 is the net rate
of vorticity shedding from the boundary layers on the upper
and lower surface of the airfoil at the trailing edge.
Then,
dr W dt dt ^^^TE^
(2.3^)
29
Fig. 2.5 The Bound Vorticity on an Airfoil
30
By calculating the circulation about a curve, as shown in
Fig.2.7, it is apparent that
dt --^TE^ dF" HF" (2.35)
So,
W dt dt + dt
(2.36)
which is equivalent tc the result obtained from th(
Kelvin-Helmholtz theorem.
An important conseq lence of boundary layer separation
may be noted by applying the dynamic free surface condition
to the wake surface exteniing from the separation point. Let
points A and B be located an infinitesimal distance ahead of
and behind the boundary layer separation point. The
pressure difference across the two points must be zero which
results in
ft ^^A - ^B^ ^ 7 ^^*A - ^^B^ = '' (2.37)
As in Eq.(2.33),
O 1
}t 1 r A 2
2 ^^^A V - ) (2.38)
Substituing into Eq.(2.39) yields
31
10
Ui
c TO •o O
X
>>
u
+>
o >
o
o +> Id
0)
•D
u c
fd i .
CO •
OJ
Hi
X
32
O
u a: a E 3
»->
4J
c •p o
Q.
«*-
O
o
c 10 X
u o o +> Id
Q: I)
X
U
C
Id I .
h-
o X
33
3(})
Tt B
3(j).
w dr
i
dF (2.39)
Therefore, it is noted that behind the boundary layer
separation point, there is an additional increment to 3(j)/3t
equal to the rate of vorticity shedding from the separation
point.
The same observation may be made by considering the
rate of change of the potential jump across the trailing
edge as described by Eq.(2.35) which may be rewritten as
9 9 u
3i 1
dr dr
at 3t + dt + dt (2.40)
For the case of a steady, stalled airfoil, the average rates
of change of ^^ and r^ are zero, yet the d(}) /dt is not zero
due to the vorticity being shed from the boundary layer
separation point.
The additional contribution to d^/dt in the separated
region is important in the calculation of the pressure
distribution around the airfoil. Without its inclusion, a
pressure jump would be indicated across the two wake
sur faces.
34
_2.2 Outline of Computer Program
The governing equations which are described in Eq.(2.8)
through (2.12) have been incorporated into a computer code
which follows the step-by-step solution method discussed in
section 2.1. The program consists of a MAIN routine which
controls the general flow of the computations and 19 su
broutines. The functions of the subroutines are as follows:
CPDIST .... calculates the airloads
DECOMP .... decomposes the influence coefficient matrix
DWSH calculates the downwash due to the free
stream
GEOM calculates the geometric description of the
airfoil
GRAD finds the surface gradient of the velocity
potential
INFO computes the influence coefficient matrix
MATRIX .... controls the matrix calculations
NUGEOM .... finds the new geometry for the wake
SETUP inputs the data and initializes the data
arrays
SHEDWK .... calculates the position of the wake
elements shed at the current time step
SOLVE .... solves the set of linear equations
TMSTEP .... controls the excution with time increren^s
35
VELWK computes the velocity vectors for the wake
vortices
WAKINF .... adds the contribution to normal downwash
due to the airfoil motion and free-stream
MOTION .... finds the location and velocity of the foil
at every time step ARCTAN .... find the
angle in the range (0.0, 2PI) whose tangent
is tan(Y/X)
SIZE calculate the viscous core size for the
wake vortices
VISCOR .... computes the local velocity induced by a
viscous wake vortex
PRTPLT .... plots the calculation results on a line
printer
Fig.2.8 is a flow chart of the calculation procedure.
36
( START
RERD THE DRTR RND SETUP THE GEOMETRY OF THE RIRFOIL
EVRLURTE THE INFLUENCE COEFFICIENTS
Yes
GENERRTE THE NEW WRKE ELEMENTS RND
FIND THEIR NEW LOCR-TION
COMPUTE THE DOWN-WRSH DUE TO THE WRKE ELEMENTS
CRLCULRTE THE DOWN-WRSH DUE TO THE
FREESTRERM
No
Yes
STOP
F i g . 2 .8 The Floiu Char t
CHAPTER III
RESULTS
The potential flow model has been evaluated for three
basic flow conditions: steady flow, unsteady nonseparated
flow, and unsteady separated flow. The calculation results
have been evaluated on the basis of comparision with experi
mental data and analytical solutions.
3_>^_ Steady Flows
Steady solutions may be obtained in two different ways:
(1) Allowing the flow to develop until the unsteady in
itial effects fade away.
(2) Forcing the initial condition to be equivalent to
the steady state condition.
Even though the first method is a viable approach to
the calculation of steady state solutions, it is computa
tionally inefficient due to the v very long computation tir.es
required. It has been found that approxi.r.ately 10 chord
37
38
lengths of travel or 100 time steps are required for an im
pulsively started rectilinear motion to approach 90% of its
steady solution. An additional 10 chord lengths are re
quired to bring the calculations to 955& of the steady va
lues. Consequently, the calculation of steady flows by this
method has been found to be impractical on the basis of the
computation time requirements.
The doublet strength distribution and circulation about
the airfoil are constant with respect to time for a steady
flow. Consequently, the rate of vorticity shedding from the
trailing edge to the wake is zero and the only vorticity in
the wake is the starting vortex which is assumed to be an
infinite distance behind the airfoil. The previously de
scribed steady potential on the airfoil may be calculated
simply by forcing the vorticity on the airfoil surface to go
to zero at the trailing edge. This is done by setting the
influence of the trailing edge vortex to zero in the formu
lation of the influence coefficient matrices. The complete
steady solution can then be obtained with a single pass
through the program.
To get a crude idea of the convergence requirements on
the number and size of the panel elements, several
calculations have been made for the circular cylinier. The
comparision of the pressure distributions for different
39
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c o •^^ •P 3 X •»-L. •P d
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numbers of doublet elements is plotted in Fig.3.1. 20 or 30
equally divided elements are noted to yield very good
agreement with the exact theoretical solutions.
To test the capability of the model to represent the
flow about a lifting body, the symmetric Joukowski airfoil
was examined. In Fig.3.2, the computed pressure
distribution around a 20 element representation of the
Joukowski airfoil is plotted against the exact solution.
The elements are spaced so as to have a denser element
distribution near the nose of the airfoil where flow
variations occur most rapidly. The element spacing has a
strong influence upon the calculated pressure distributions.
This is especially true around the nose of the airfoil.
The pressure distribution for the NACA0015 airfoil is
tested in the same fashion as with the Joukowski airfoil,
with the results illustrated in Figures 3.3 through 3.6.
3.2 Unsteady Flows
The case of an impulsively started flat plate airfoil
provides a useful test of the potential flow model in an un
steady flow. Fig.3.7 exhibits the numerically calculated
distribution of potential jump along the airfoil at the
starting instant together with the exact solution. Fig.3.?
46
illustrates the indicial circulation and lift experienced by
the flat plate during the impulsively started rectilinear
motion. Comparision is made with the linearized analytical
solution. The presented flat plate computation uses 10
equally spaced elements and employs a time step equivalent
to the time needed for the airfoil to travel 0.1 chord
lengths or the length of the trailing edge element. This
follows the suggestions of R. H. Djojodihardjo and
S. E. WidnalK1969) . Fig.3.9 shows the instanteneous wake
geometry after 0.5 and 1.0 chord lengths of the travel. The
beginning of the rollup of the wake is observed after 1
chord length of tavel in Fig.3.9-
To illustrate the realistic manner in which the wake
geometry may be predicted, the calculated wake behind a
peoridically plunging flat plate airfoil is presented in
Fig.3.10.
The pressure distribution on an implusively started
NACA0015 airfoil, after 1 and 5 chord lengths of travel, is
illustrated in Fig.3.11- It can be observed in Fig.3-11
that the pressure gradually changes its distribution to the
steady state condition.
47
^0
1.0 r
.5
.5 1.0 X/c
Fig. 3-7 Potenttal Jump Distribution on an Impulsively Started Flat Plate
48
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52
2*2. Separated Flow
When the boundary layer separates prior to the
trailing edge, an additional wake surface must be generated.
Figures 3«12 and 3»13 illustrate a typical calculated geome
try for the wake in a separated flow and the type of diffi-
culies which have been encountered in making these calcula
tions. From the figure, it is apparent that the wake
surface has been allowed to across the airfoil surface.
This is a consequence of the combined effects of modeling
the airfoil and wake surfaces with discrete vortices and
representing an unsteady flow with a step-by-step solution
method utilizing finite time increments.
Fig.3.14 depicts the variation of lift and drag with
respect to angle of attack. For these calculations, it was
necessary to assume a location for the boundary layer sepa
ration point as illustrated in Fig.3-15. The data points
for the separated flow were taken from the time step previ
ous to the event of the wake crossing the airfoil surface.
Several modifications to the potential flow model are
currently being evaluated. These changes are directed at
solving the wake crossing problem and improving the overall
accuracy of the model .
53
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55
The wake's crossing of the airfoil surface is primarily
the consequence of the descritization of a continuous fluid
flow problem. For instance, the airfoil surface is modeled
by a finite set of doublet panels whose strength are
determined such that the flow is required to be tangent to
the airfoil surface at a finite number of control points.
Between the control points, the flow is not tangent to the
surface and the streamlines go in and out of the airfoil at
various points. A free wake vortex could be convected
through the airfoil at critical points between the control
points. Possible remedies to this particular problem are to
use more elements to model the airfoil surface or to use
higher order doublet distributions on the elements so that
the number of control points can be increased. Schemes
involving the interpolation of velocities between control
points are also being considered.
Recall that the wake surfaces are modeled by discrete,
potential vortices, which induce infinite velocities as
their centers are approached. As a consequence,
unrealistically high induced velocities are indicated when
vortices happen to be convected very close to one another.
This problem may be alleviated by utilizing Rankine vortices
which have viscous cores instead of the inviscid, ootenUal
vortices (Fig. 3.16). By reducing the induced velocity and
56
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58
e c o u tA 3 O O (A
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59
consequently the distance traveled by a vortex during a
finite time step, the chance of its crossing the airfoil
surface is reduced.
At the current time, the flow near the boundary layer
separation point is only crudely modeled. Ideally, a Kutta
condition similar to that enforced at the trailing edge
should be utilized at the separation point in the linear
equation set for the solution at any instant in time. This
would guarantee a stagnation point at the boundary layer
separation point. Instead, vortices are injected just above
the airfoil surface at each instant in time which have the
effect of retarding the flow in that region. A better
representation is the generation of a uniform strength
vortex element from the boundary layer separation point at
each time step. The total vorticity in the elements are
lumped into discrete vortices as they are convected away
from the surface and new elements are created.
CHAPTER IV
CONCLUSIONS AND RECOMMENDATIONS
The two-dimensional airfoil model described in this
thesis is theoretically capable of the prediction of air
loads for both nonseparated and separated flows. However,
the calculation results reveal that, in its present form,
the model is only partially successful. On the basis of
those calulations, the following conclusions and recommenda
tions may be made:
1. The model does a satisfactory job of calculating
steady and unsteady nonseparated flows.
2. Accurate predictions of the airloads on airfoils with
boundary layer separation can not be made until sig
nificant modifications of the model are implemented.
3. One refinement needed in the model is a more precise
representation of the flow in the vicinity of the
boundary layer separation point.
4. In addition, the airfoil surface should be modeled
differently so that the velocity along the surface is
more continuous than the present model allows.
60
REFERENCES
1 .
2.
3.
4.
5.
6.
7.
8.
9.
10
Deffenbaugh, F. D. and Marshall, F. J. "Time Development of the Flow about an Impulsively Started Cylinder," AIAA J., Vol. 14, July 1976, pp. 908 - 913-
Djojodihardjo, R. H. and Widaal, S. E. "A Numerical Method for the Calculation of Nonlinear, Unsteady Lifting Potential Flow Problems," AIAA J., Vol. 7, 1969, pp. 2001 - 2009.
Gerrard, J. H. "Numerical Computation of the Magnitude and Frequency of the Lift on a Circular Cylinder," Philosophical Transaction of the Royal Society of London, Vol. 261, Jan. 1967, pp. 137 - 162.
Ham, N. D. "Aerodynamic Loading on a Two-Dimensional Airfoil During Dynamic Stall," AIAA J., Vol. 6, 1968, pp. 1927 - 1934.
Karamcheti, K. "Principles of Ideal-Fluid Aerodynamics," John Wiley and Sons, 1966
Katz, J. "A Discrete Vortex Method for the Non-Steady Separated Flow Over an Airfoil," J. Fluid Mech, Vol. 102, 1981, pp. 315 - 328.
Lamb, Horace "Hydrodynamics," Dover Publications, 1932
Kraus, W. "Panel Methods in Aerodynamics," Numerical Methods in Fluid Dynamics, McGraw-Hill, Editors Wirz and Smolderen, pp. 237 - 297.
McCroskey, W. J. "Inviscid Flowfield of an Unsteady Airfoil," AIAA J., Vol. 11, 1973, PP• 1130 - 1137.
McCroskey, W. J. and Phillippe, J. J. "Unsteady Viscous Flow on Oscillating Airfoils," AIAA J., Vol. 13, 1975, pp. 71 - 79.
61
62
11. Oler, J. W. "An Investigation of a Numerical Method for the Exact Calculation of Unsteady Airloads Associated with Wing Intersection Problems," M.S. Thesis, University of Texas - Austin., 1976
12. Oler, J. W. & Strickland, J. H. "Dynamic Stall Regulation of the Darrieus Turbine," Progress Report on Sandia Contract No. 7^-1218, 1980, pp. 87 - 99
13. Sarpkaya, Turgut "An Inviscid Model of Two Dimensional Vortex Shedding for Transient and Asymtotically Steady Separated Flow Over an Inclined Plate," J. Fluid Mech., Vol. 68, part 1, 1975, pp. 109 - 128.
14. Sarpkaya, T., and Schoaff, R. L. "Inviscid Model of Two-Dimensional Vortex Shedding by a Circular Cylinder," AIAA J., Vol. 17(11), 1979, PP• 1193 - 1200.
15. Summa, J. M. "Potential Flow About Impulsively Started Rotors," J. Aircraft, Vol. 13, 1976, pp. 317 - 319.
