Non-local technique for
nonlinear boundary value problems in supersonic flow model
Ricardo T.Ferreyra, José P.Tamagno
Universidad Nacional de Córdoba
Córdoba, Argentina, CP. 5000
Nomenclature
V1 - velocity of the current flow, m/s
M1 - Mach number of the current flow, dimensionless
qc - solid cone semi-angle, radians
qs - shock wave semi-angle, radians
q - intermediate angle between the cone and the shock wave, radians
VLím - highest dimensional velocity that the h0 enthalpy flow can reach, m/s
VR - component of velocity in the radial direction, m/s
Vq - component of velocity in the tangential direction, m/s
ps - pressure at shock wave, N/m2
p02 - static pressure after the shock wave, N/m2
ps /p02 - pressure ratio, dimensionless
(×)c - velocity or pressure properties over the solid cone
(×)s - velocity or pressure properties over the shock wave
Ms - Mach number at the shock wave, dimensionless
Cp, Cv - specific heat at constant pressure and at constant volume, J/K
g - ratio of specific heats, dimensionless
a - velocity of sound, m/s
h0 - constant enthalpy, m2s2
d - dislocation function, dimensionless
e - internal energy per unit mass, J/kg
V - velocity vector, m/s
r - density function, kg/m3
P - pressure function, N/m2
- rate of heat added per unit of
mass, J/kg s
f - body force per unit of mass, N/kg
s - entropy function, J/K
h - enthalpy, m2/s2
1. Introduction
The development of an approximate analytical solution
to the Taylor-Maccoll boundary value problem that describes the inviscid
supersonic flow past a solid cone at zero incidences is described. The
Taylor-Maccoll formulation results in a second-order nonlinear ordinary
differential equation, [1], whose associated numerical solution has existed
since 1933. Numerical solutions to the equation are easily obtained, which are
currently accessible via MATLAB, [2-3]. Tables and charts presenting solutions
to the Taylor-Maccoll have been available for many decades. However, analytical
methods are preferred when the purpose is to know explicitly relationships
between the physical variables (namely 
, 
and 
).
Over recent decades, there have been numerous approximate analytical representations, for the analysis of this and other closely related problems, which have usually been presented as perturbations or series expansions, [4-8]. However, to date, no closed form solution to the Taylor-Maccoll equation has been presented, [8-14].
In the present work, a solution was formulated to
solve the original Taylor-Maccoll problem for a conical body of circular cross
section at a zero angle of attack through an analytically closed form family of
solutions (not expansions) within the interval < 
< . The
procedure, which is self-contained, was developed as a complementary tool for
exploration techniques to solve partial differential equations.
As a result, it is now possible to determine
analytical knowledge of radial velocities 
, 
, 
, tangential velocities 
, 
, 
and pressure
ratios 
,
, 
in the
physical region < < . Finally, it should
be noted that, to our knowledge, the basic Cone solution, CS,
here proposed, cannot be found among existing solutions.
2. Theoretical review
The differential equations of momentum, continuity and energy for inviscid flows and the definition of enthalpy have the following form:
(1)
(2)
(3)
The flow is adopted with zero vortex, thus 
. Also, the
flow is assumed to be adiabatic (
), with no body forces (f = 0),
steady (
),
and isentropic (Ds=0). The relationship between pressure and
density is adopted in the form 
, where k is a constant value and g is the ratio of specific heats. The
velocity of the sound satisfies 
. Therefore, the momentum, continuity and
energy equations can be rewritten as:
(4)
(5)
(6)
Crocco's theorem establishes that 
. Now,
since, 
,
and 
, then 
. Equation (6) implies
a level of enthalpy called h0, defined throughout the kinetic
energy as 
and
associated to a fixed reservoir condition. In fact, when this condition is
reached the flow has expanded to zero temperature, and hence h=0.
(7)
Note that Vlim is a constant for the flow obtained from h=0, and the equation. Vlim is the highest dimensional velocity that the flow can reach under the hypothesis of a constant enthalpy expansion process with a value of h0.
Combining the momentum and continuity equations (4) and (5), and considering a calorically perfect gas, the equations (4), (5), and (7) become:
(8)
(9)
For a potential function f such that 
exists, then the vector velocity in
spherical coordinates has the form 
, and the hypothesis of conical flow is
established as 
. If axis-symmetric flow exists, then the
derivatives with respect to w must
be zero, 
,
and consequently 
. This implies that 
, 
, 
and 
, and that
the flow properties are constant along a ray from the vertex 
. Since the
non-dimensional Mach number is 
, then from (9):
(10)
From (8) and (9), and using the non-dimensional velocity V=V/Vlim, the dimensionless equation of Taylor-Maccoll can be written in the following form:
(11)
The unknown terms of equation (11) are the
radial and transversal components of velocity. The transversal component, 
, is
considered null on the surface 
. Therefore,
(12)
The shock boundary problem still remains unsolved. Consequently,
the boundary conditions associated with the conic shock angle 
must be
imposed. The radial component of the velocity, 
, is tangent to the shock wave
and thus,
(13)
and the transversal component at the shock wave, 
, must
satisfy the equation:
(14)
Equations (10) and (11) come from typical physical and
geometrical considerations at the shock wave, and they are usually referred to
as the Rankine-Hugoniot shock conditions. The symbol 
represents the free
stream velocity, while the symbol 
is the corresponding Mach number. The
set of equations (11), (12), (13) and (14) define the boundary value problem to
be considered. This paper aims to find an approximate analytical solution for
this set of equations.
3. An analytical dislocation method
Suppose that
(15)
is a nonlinear differential equation and 
,..,
, are the set
of boundary conditions associated to the physical problem. If exists
such that
, and 
, then S
is the exact solution of the boundary value problem. If f and S
are analytic, then it is always possible to write 
in the form of an expansion 
. Also, if 
then,
, and 
. It
follows, 
or
simply the form
(16)
It is said that a dislocation happens when something
leaves its original location. So, d
is called the dislocation function and 
is called the dislocated term. The form
of Eq. (16) is chosen depending on the purpose. For instance, in searching for
the roots of 
, the option 
, where 
, 
, and 
, shows that
the first root of 
is 
. On the other hand, 
, where 
, 
, and , shows that
the second root of is 
. In consequence, the roots of 
were found.
In general, 
is simpler than 
. Also, if q0 exists, then (in the limit q = q0) both equations 
and 
are
satisfied. Therefore, if 
or 
, then q0 is not the solution. Furthermore, if 
, then the
solution does not exist. For instance, 
, where 
has roots 
, 
, and shows that 
. In fact, 
and 
.
However, sometimes it is difficult to find
a closed form solution, not expansion, even when the exact solution exists. Frequently,
additional information of the equation is available. The idea is to achieve the
reverse order. That is, to propose a priori the equivalent form 
, and to
solve for 
,
whenever 
is
addressed. In this new context, a local dislocation implies that some terms of can be
dislocated from their original places. Since has a finite number of terms,
perhaps two or more, then it can be rearranged in the form 
. The set of terms 
can be
isolated in order to form a new equation comprised of the remaining terms. The
new equation 
may have an analytical
accessible solution 
such that
(17)
On the other hand, it is assumed that the composite
function 
is
analytic and exists 
such that 
. Since 
is also an analytic
function, then the zero is isolated and exists 
such that 
, where 
, and exists
a neighborhood of 
in which the condition 
is true.
Notice that may or may not exists one (or maybe more) isolated points 
in which 
at the time
that
.
To summarize, it has been shown that may have the
form 
or
whenever
is
satisfied. In addition, a limiting process is addressed: if 
(or 
), then 
(or 
), 
, 
, and the
desired condition 
is reached. Notice that it has been
demonstrated that 
exits. However, how to find for any
specific case depends on the problem.
Finally, it should be pointed out that in
the beginning of the dislocation process some terms 
can be added to in a
balanced form 
and rearranged to obtain 
in a way
that facilitates finding an exact solution 
for 
. Therefore, 
.
These general rules have no restrictions. They are
also applicable when the analytical estimation 
is obtained rather than a
closed form solution. Thus, this practice allows an approximation to be
reached. However, if 
(the solution does not exist), then it
is probably that should not be applied in solving
problems of nonlinear analysis in engineering systems. Nevertheless, what can
be difficult is to know when the solution S(q) cannot exist. That is 
. Although in the case that
there is no solution, the dislocation technique should have sense. In fact, an
approximation 
can be computationally, geometrically or
experimentally checked within the context of an engineering analysis.
As a first example, note that the form (15) and can be replaced by Eq. (16). Also, suppose f satisfies the limiting process
(18)
Once 
is proposed and Eq. (18) is addressed,
it does not matter who is .
For instance, consider the specific function cos(q), whose roots 
are well known, and
four boundary conditions 
, 
, 
, and 
. Suppose that, 
, 
, q0=0, and d(q) =q 7 because q 7 cannot be
approximated by 
. According to LDM, the four boundary
conditions should be rewritten in the form 
,
, 
, and 
. These boundary
conditions are necessary to obtain 
.
Finally, the interval 
contains the points in which
the boundary conditions have been imposed, and is the same interval where 
. Note that
the root q =1.6986 approximates q =p/2.
As a second example, it is possible to demonstrate that any perturbation scheme can be considered as a particular case of this methodology. Let us consider the equality T×t =V and expand the factors in the following way:
(19)
The symbol e is a small parameter. After arranging terms and then carefully selecting every part of the scheme as:
(20)
then a dislocation exists. In fact, if d = e, 
, and 
, then the
equation has the typical form
(21)
The assumption d = e n is also
possible after arranging most of the terms that belong to f *
and f **. Thus, this perturbation scheme can always be
arranged in the form of a LDM scheme, and the solution of f *
= 0 can always be approximated by a perturbation scheme, [15]. For nonlinear
problems, this is probably a realistic expectation. Notice that if d R 0, then every part of Eq. (21) verifies the stated
LDM requirements. To illustrate this idea, rather than solving the model (as
this is given in the literature, [15]), let us consider a dynamical system 
and 
, where 
is a real
number, x is the position and 
is the second derivative of the position
with respect to t. Assuming 
to be a solution, then the approximate
global model is 
, and its solution has the form 
, where a0
and b0, amplitude and phase, respectively are assumed to
depend on e. Thus, the value of 
depends on e, and e is a link to higher order terms. In the case of LDM,
the expression (not only the value) of 
depends on the formulation of 
, and the
formulation of 
results once is chosen, or
reciprocally. Finally,
.
In this generic example, LDM does not replace what any other particular method does. However, LDM reformulates an original global model (or non-local problem) without a solution into an approximate global model (approximate non-local problem) for which the solution is known as a closed form solution or as a perturbation solution.
To conclude, LDM can facilitate the process of:
1) building an
approximate analytical model, 
,
2) finding a
closed form solution 
, for which the solution is numerically
known (or any form of solution is known). To this end, let us consider as a
third example the Taylor-Maccoll boundary value problem for which the
approximate solution is obtained in the next section.
4. LDM applied to solve a Taylor-Maccoll model
The Compact form of the Taylor-Maccoll equation has the form:
(22)
The factors T and t of the left member are given by:
(23)
and 
(24)
and the right member is:
(25)
The analytic functions 
and 
both satisfy the
Taylor-Maccoll equation. However, the fact that these functions do not satisfy
the Taylor-Maccoll boundary value problem (as this is given above) is essential
to the analysis. The availability of undetermined constants in the solution is
necessary due to the fact that some arbitrary constants must be consumed by the
restrictions. An analytic solution 
with the three undetermined constants 
is
sufficient since there are three boundary conditions to be satisfied. To
facilitate the procedure, the condition 
must receive early consideration. Thus,
an analytical approximation, 
, for the boundary value problem is
proposed as:
(26)
where 
still has to be determined. By replacing
the proposed solution, Eq. (26), into the original equation 
, and by adding,
removing and resetting terms from the equation , it is easy to construct an
equivalent expression in the form:
(27)
which satisfies the four LDM requirements
described above. Once has been identified, Eq. (26), then LDM
provides the scheme, Eq. (27). Therefore, the sets of terms T*,
V*, T**, V** and d are respectively:
(28)
(29)
(30)
(31)
(32)
Equation (26) is a closed form solution of 
. The proof
of this is given in Appendix 1. The flexibility in finding an analytical
relationship is one of the most attractive features of the solution technique
developed in this paper. Furthermore, Eq. (26) is quite simple to use and can
be easily solved by hand, even though is a non-linear partial differential
equation. In addition, if the set of dislocated terms 
is considered to be
the error, then replacing CS in the dislocated terms helps to estimate this
error:
(33)
Equation (33) represents a continuous function, which
must satisfy 
when an additional condition is imposed,
such as 
R 0 or d R 0.
The approximate trigonometric solution 
given in Eq.
(26) is referred to here as the Conic Solution or CS solution, because of three
illuminating reasons. First, it immediately satisfies the boundary condition, 
, over the
cone surface. Second, the factors d,
, are easily
interpreted using CS. Third, d, as a part of the dimensionless total enthalpy 
, can be
rediscovered once it has been verified that 
. Then, as:
(34)
Hence, 
implies that 
if and only if d R 0. However, if d must satisfy d ¹ 0 in the problem, then 
should be imposed. As the
involved functions are continuous, the proposition 
implies 
. Thus,
either or
is
sufficient to ensure that 
within the dislocation scheme 
.
In other words, the CS solution can be interpreted as
the "key" in transforming the Taylor-Maccoll equation 
through the
degenerative process , in which 
.
The radial velocity at the shock wave, Eq. (13),
imposed on the CS solution Eq. (26), implies 
. Thus,
(35)
Eq. (26) implies that if 
, then 
. On the other hand,
Eq. (26) and Eq. (35) imply that, if , then 
. If Eq. (32) is valid, then,
the estimation given by Eq. (26) also provides an exact solution for the
following boundary value problem:
(36)
which is an approximate model for the full problem. Thus, at this point, the method has provided a closed analytical solution, Eq. (26), which is an exact solution for Eq. (33), but is an undetermined approximate solution to the full problem. There is also the requirement that all boundary conditions must be met. To this end, Eq. (14), associated to the transversal velocity at shock, is imposed at the time a relationship for qS, qC and V1 is obtained. In this process, the differentiation of Eq. (26) provides a new formulation for the transversal velocity:
(37)
Also, a new relationship is attained by combining equations (13), (14), (26), (32) and (35) at the shock:
(38)
Equation (38) summarizes the whole model, since it has been obtained by considering all the boundary conditions and the proposed solution. It can also be explicitly solved for qC by considering qS and M1 as independent variables:
(39)
Although it is normally difficult to find an explicit relationship in solving non-linear differential equations, even in the approximate case, here the goal of reaching an analytical explicit relationship for the full boundary problem has been easily attained. This feature justifies the dislocation resource.
An additional explicit form arises by solving for the
absolute value of the Mach of the free stream, 
, as a function of both qC and qS:
(40)
Finally, in solving for qS, Eq. (40) shows that qS is an implicit function which depends on 
. However,
the graph of 
is the same set of points as the graph
of 
,
so Eq. (40) becomes useful.
Some level sets for 
are provided in this paper to
produce different curves in the 
plane. It should be noted that as there
is a fractional exponent in Eq. (40), then there are two mathematical branches
for the functions
and 
. However, perhaps only one
branch has any physical meaning or interest.
Widely used boundary value problem ODE solvers are able to provide solutions almost instantaneously, but they require specialized software or programmable hardware. On the other hand, Eq. (40) (not Eq. (39)) allows specialists to obtain charts with the simplest scientific pocket calculator. Finally, in contrast with computer-numerical codes, CS provides an explicit relationship between internal variables.
5. Results
The proposed Local Dislocation Method has been applied to provide an approximate solution to the full Taylor-Maccoll model. Some physical and analytical insights given previously to justify the local validity of the results in the region where the convergence takes place are now discussed.
A. The pivotal relationships 
For comparative purposes, it is of interest to
estimate the error incurred when the dislocation method presented is used.
Therefore, curves of the analytical errors,
, as a function of for
different 
values,
are developed:
(41)
If , then Eq. (41) becomes:
(42)
The conic solution CS exactly solves 
. Thus, 
. Then, it
follows
that: 
(43)
Then, the error,
, is associated with the applied
methodology and depends on 
at every point. Results from Eq. (43)
illustrate the importance of the ratio between omitted terms and non-omitted
terms when LDM is applied; see Fig. 1, with the error obviously depending on
the omitted terms. From Eq.(43) 
implies 
. Furthermore, 
.
Figure 1 shows that the relative error
increases dramatically with increasing free stream Mach number for 
, with curves
obeying a nonlinear law. The value qC=25 is
estimated from the set of curves. Notice that, if qC£25, then the curves are
virtual straight lines with very little slopes. Thus, for all 
values, the
largest error depends on a fixed value of 
. For the whole M1
interval, as decreases the largest error also
decreases. The criterion 
was taken as the procedure to follow in
order to reach a good approximation in the analysis of the error, with this
behavior being shown in Fig. 1. Consequently, the geometrical condition restricts
the validity of the CS approximation to a sector in the chart. The value qC=20 was adopted as the
maximum value for which the assumption 
was nearly valid. This is observed by
the fact that as increases, the curve associated with qC=20 is the last
approximately linear one represented in Fig. 1. After that, curves are
non-linear and the error 
starts increasing.
At this point, it
is clear that implies 
. In addition, 
signifies
that 
.
However, ,
does not imply aR0 while a
variation of geometry of the cone is taking place (). The proposition 
is not necessary, as it
can be seen that 
is
sufficient for applying LDM.
In this context, the dislocation technique can be evaluated by comparing the curves obtained from the conic solution CS against the Taylor-Maccoll results, which are considered to be the reference solution.
In our research it was analyzed:
B. The limit 
C. The region 10£qC£20
D. Surface Mach numbers and pressures in the whole region 0£qC£40.
(But the corresponding figures and formulas are not shown here).
6. Conclusions
A non-trivial dislocation process, LDM, is proposed and has been successfully applied to obtain a closed form solution for an approximate Taylor-Maccoll boundary value problem suitable for a supersonic cone at zero angle of attack. As a result, one of the analytic outputs of the LDM method has been named the conic solution, or CS solution.
Results show that the analytical solution is in
agreement with the Taylor-Maccoll solution for the half cone angle range of 
. Inside,
three different parts of the convergence region, 
, 
and 
, have been
characterized, and an analytical approach for pressures and surface Mach
numbers provided.
The CS analytical solution is an analytical tool that can be used to extrapolate results in the physical region to solve the flow field between the real supersonic cone and the shockwave.
Appendix 1
This Appendix includes the necessary guidelines to perform the proof of the Proposition
Consider the following relation, named the conic solution, CS:
(A1-1)
and also the equation 
(A1-2)
which is explicitly defined by the involved factors (A1-3), (A1-4) and (A1-5)
(A1-3)
(A1-4)
(A1-5)
Then, the conic solution, (A1-1), is an exact analytic solution of (A1-2).
Proof
If the left member of equation (A1-2) is written using (A1-3) and (A1-4) and evaluated with the conic solution, (A1-1) then
(A1-6)
In the same way, if the right member of equation (A1-2) is written with (A1-5) and evaluated with the conic solution (A1-1), then
(A1-7)
Therefore, from (A1-6) and (A1-7), the equality (A1-2) then becomes straightforward. Hence, (A1-1) is an analytic exact solution of (A1-2).
Acknowledgments
The first author acknowledges the academic and research national program, PROMEI, Ministerio de Cultura y Educación de la República Argentina, for the financial post-doctoral support. The authors thank Dr. Paul Hobson, native speaker, for the revision of the manuscript.
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