Active control of the flow using various methods
of influence, including the release of energy in the flow field [1-7], is an
urgent problem, which is still quite far from a satisfactory solution. One of
the areas of application is the control of flows in which the interaction of
shocks of
the Edney
classification is realized [8,9].
When oblique and direct shock waves intersect, there are six different types of
interaction [8], which differ greatly both in the flow structure and in the
values of pressure and heat flows on the surface of the body. Moreover, the
most intense effect on the surface of the body is realized during an
interaction
of the
Edney type.
With this type of flow, a narrow stream is
formed, which is decelerated in several successive oblique shock waves and
reaches the surface of the body with small losses of total pressure. This leads
to very high values of pressure and heat flow at the surface of the body.
In this regard, it is
of practical interest to be able to actively control the flow in order to
destroy the flow structure of a type
and replace it with a structure that is less dangerous,
from the point of view of loads. In [3, 4], the destruction of this flow structure
using pulsed heating was experimentally studied. In [5], numerical experiments
demonstrated the possibility of reducing the maximum pressure and heat fluxes
on the surface of a body by releasing thermal energy in the vicinity of the
incident shock. Calculations have demonstrated the possibility of reducing the
maximum pressure and heat flows on the surface of the body, however, the work
indicates that the control action is not optimal in magnitude. One of the main
reasons preventing the practical application of flow control methods is their
fairly high energy intensity, so the question of finding the minimum impact in
terms of integral value is of significant interest. Another obstacle is the
high complexity of searching for zones and magnitudes of influence, so
efficient computational algorithms for searching for control actions are also
of interest.
In this paper, within the
framework of a two-dimensional model of an inviscid gas, a method and results
of searching for spatially distributed stationary heat sources of minimum total
power that reduce the maximum value of pressure on the surface of the body are
presented. The main tool used is an iterative solution of the inverse problem
of minimizing a certain functional using gradient methods. The gradient of the
functional with respect to the sources was calculated using adjoint equations.
Let us consider the flow of a supersonic
flow around a flat body with a cylindrical bluntness when an inclined shock
wave falls on it. We use non-stationary Euler equations:
here
,
,
,
,
.
The calculation area is rectangular
.
The time interval
was
selected from the conditions for establishing a stationary regime.
A spatially homogeneous field was taken as the initial conditions.
At the inlet boundary we accept inflow conditions
corresponding to an inclined shock wave in a homogeneous flow.
We set
outflow conditions at the output boundary and the lateral boundaries of the
region. No-flow conditions are specified at the body boundary.
Euler's equations were solved using second-order
reconstruction [11], methods [12] and [13] for solving the Riemann problem, and
the fourth-order method [14]. For the second-order method, it was possible to
reproduce a
type circuit on a grid
,
for a 4th-order accurate circuit on a grid
.
Boundary conditions on the body are implemented by the immersed boundary method
[15-17].
Figure 1 shows density isolines
for a type
IV configuration,
corresponding to the
operating conditions [5] and
obtained in
calculations
on
a grid
using
[11,13].
Flow parameters:
,
flow deflection angle behind the shock
,
.
Figure 2 shows the pressure distribution in the flow
field in three-dimensional form.
As the oblique shock moved upward along the
coordinate,
a sequential transition from a
type
scheme through a
type scheme to a
type scheme was observed. In this case, the flow
according to
the scheme
has
varieties that differ significantly in pressure on the body [19]. At low
pressures
(at the border with the circuit
)
the pressure is maximum, then, as it increases
,
a second maximum is observed; finally, the circuit
turns
into a circuit
in which the high-pressure jet bends upward and does
not hit the body, which leads to a decrease in the maximum pressure.
Figure 1. Density
isolines for
the type of interaction
Figure 2. Pressure distribution
for
the type of interaction
There are several possible scenarios for the
transition from a type
IV
configuration to more favorable structures.
In [5], due to the heat source, the shock wave
is deflected, leading to a transformation of the
type flow pattern to the
type pattern. In [1], a complete restructuring of the
flow with the formation of a forward separation zone (“thermal needle”) is
considered. Both of these options involve the use of a localized point source.
Here we will consider a spatially distributed field of thermal sources.
Let us consider a steady flow with an interaction
of the type, Fig. 1.2.
We need to select a heat flow distribution
that
changes the flow pattern so as to minimize the maximum static pressure on the
surface of the body:
|
 ,
|
(4)
|
(
- coordinates of the point of maximum pressure on the body).
At the same time, the integral power of
thermal sources
|
|
(5)
|
(taking into account the positivity of sources,
corresponding to the norm
)
should be minimal.
Unlike works [5-7], in this work the control is
carried out not with the help of an initial temperature disturbance, simulating
a pulsed heat release, but with the help of time-constant heat sources.
The work also used
functionals like:
Increasing the indicator
in the expression
,
makes the functionality more
sensitive to changes in pressure and, generally speaking, allows you to more
accurately identify zones of maximum pressure, which is due to the fact that
.
In expression (6), the zero-order Tikhonov
regularization is used to ensure the search for the minimum disturbance. The
regularization coefficient
is selected during calculations.
To
calculate
the gradient of the
target functional,
conjugate equations of the form [6,10] were used.
|
|
(7)
|
|
|
(8)
|
|
|
(9)
|
|
initial conditions
|
(10)
|
Boundary conditions (at
the boundary of the computational domain):
.
The boundary
conditions (on the body) for
functional
(4) have the form
|
 ,
|
(11)
|
The boundary
conditions (on the body) for
functional
(6) have the form
|
 ,
|
(12)
|
The form of the
functional (4) leads to boundary conditions (11) containing
the
function on the boundary. In
the
case of movement of the zone of maximum pressure along the surface of the body,
this shape
can lead to sharp changes in the
associated parameters and the corresponding gradient
and complicate the use of gradient methods.
Therefore, the
main part of the calculations was carried out using form (6) and boundary
conditions (12).
The conjugate equations were solved using the
numerical method in [18] similarly to
[6,7,20].
Using the
results of solving the conjugate equations, one can obtain
the radient of the
target functional of the following form:
|
|
(13)
|
The search for optimal control is carried out by an
iterative method using the steepest descent method
|
|
(14)
|
The results of gradient calculations for
and for
and
are
practically the same in shape (up
to scale), which makes them equivalent when using gradient optimization
methods. This is probably due to the presence of a sharp pressure maximum for
this task and will not be fulfilled for tasks with a weakly expressed maximum.
The distribution of the gradient value (13) in the
calculated field is presented in Figure 3 in three-dimensional form.
Figure 3.
Gradient of the target functional
in
the
interaction mode
It is possible to distinguish three zones in which
heating will reduce the pressure on the surface and one which will increase the
pressure.
One of the zones of maximum
gradient coincides with the incident shock, which corresponds to the mechanism
of weakening the impact due to the deflection of the incident wave. The second
corresponds to the entry of the streamline strictly into the region of the
high-pressure jet, the third (minimal) corresponds to the entry of a
disturbance of the opposite family relative to the falling shock.
From Fig. 3 it can be seen that the gradient in different
zones has different signs, which during iterations can lead to a negative
density of heat sources, which is physically not realizable. In these
calculations, the density of heat sources was projected in the positive
direction. Each iteration corresponds to solving one direct and one conjugate
problem.
The regularizing coefficient plays a
special role here
a.
At a small value of the regularization
coefficient,
heat sources completely distort the flow field. At
power of thermal sources
(which
is significantly more than in [5]) and
.
The results were obtained in 9 iterations. The flow field has undergone
significant restructuring; the corresponding field of heat sources and pressure
are presented in Fig. 4, 5.
Figure 4. Heat source density
,
Figure 5. Isolines of the pressure field
,
Increasing the regularization
coefficient makes it possible to reduce the integral intensity of the sources
by reducing the control efficiency (increasing the maximum pressure); the
corresponding results for
,
(
,
)
are presented in Fig. 6, 7.
Figure 6. Heat source density,
,
Figure 7. Isolines of the pressure field, (
(),
The results were obtained in 4 iterations. It should
be noted that the flow field has undergone a restructuring corresponding to the
transition to the type of interaction
[19].
Calculations demonstrate the
possibility of effectively searching for control actions and the significant
influence of regularization. The results of solving the inverse problem
demonstrate the transformation of the flow not according to the “shock
refraction” scenario
with the transition from
structure
to
[5],
but
the transition
from interaction
of type to
structure of type
.
In the work of Kogan and Starodubtsev [5],
a heat source of the following form was used:
For flow parameters
,
(flow deflection angle), the results of [5]
are generally reproduced, and it is possible to reduce the maximum pressure
value from
to
.
The required
power
of heat sources
.
As another option for controlling the flow field, the
“thermal needle” mode is implemented [1]. The “thermal needle” mode is
presented in Fig. 8 using pressure field isolines. At the same
time
.
Figure 8. Isolines of
the pressure field
during
separated flow induced by a “thermal needle”
The search for the
distribution of heat flows that optimally reduces the pressure on the body
strongly depends on the regularization coefficient and leads to different flow
structures.
The transition of
the flow from a structure
to
a “thermal needle” [1] requires a radical
restructuring of the flow, which comes down to a significant change in the
effective shape of the streamlined body,
The flow transition
from the structure
to
the shock using local refraction [5] is also
quite energy-consuming even if the heat release zone successfully hits the
shock wave.
In this regard, the
management of the transition from the interaction
of the type purposefully to the structure of
the type
described in
[19] is considered. This type of structure
is characterized by the fact that the
high-pressure jet bends along the body and does not fall on it.
In this regard, in
this work we used the following target functional:
|
|
(16)
|
here is
a region in the calculation field
containing some part of the flow structure
interactions,
- flow parameters corresponding to
the structure
.
These parameters are taken from
calculations for
type of shock interaction obtained by
moving a falling shock. Thus, we purposefully strive to reproduce the known and
convenient flow structure in a certain area
.
Calculation results for the following
functional are presented
|
|
(17)
|
In Fig. 9 and 10 the frame highlights the area of
observation and corresponding adjustment of the flow
.
Figure 9. Target current (isolines
)
type
Figure 10. Reproducing the target flow in the window
(isolines
).
To completely match the fields in
,
but the residual functional decreases by an order of magnitude.
Figure 11 shows the location of heat sources.
It was possible to reduce the
maximum pressure by half while consuming energy
.
Figure 11. Density of a heat source during a targeted
transition to flow type
Thus, reproducing a certain “target” flow that
has the desired properties is also possible by controlling parameters on the
body.
Generally speaking, it
is quite easy to find
heat sources capable of destroying a low-entropy jet, which represents the main
danger from the point of view of high values of static pressure on the surface.
This is due to the relatively narrow range of parameters within which
the genus interaction exists. More difficult is the
search for an impact that is minimal in terms of the total
power of heat sources.
Three different options for choosing a control action
are considered. The first uses a “thermal needle” [1] and completely rebuilds
the flow field by forming a front separation zone.
The second is based on the refraction of the shock
wave in the heated region and the displacement of the intersection point of the
incident and direct shocks, which transforms the interaction from type
to
type
in accordance with the results of [5].
The third is based on an automatic search for the
distribution of heat sources by solving the inverse problem.
As part of solving the inverse problem, the minimization of
the control action norm is carried out using a regularizing addition to the
objective functional.
The maximum surface
pressure and energy consumption depending on the regularization coefficient are
given in Table 1.
Table 1.
Regularization makes
it possible to reduce the integral power of the heat source below the results
[5] (where
),
but
at the same time increases the maximum pressure, which is natural when moving
from a “thermal needle” type structure containing a separation zone. The search
for minimal control by solving the inverse problem is complicated by the search
for the optimal regularization coefficient. Purposeful transformation of flow
from type
to type
is the simplest from the point of
view of the cost of computing resources.
For low powers of
heat sources, the change in pressure on the surface is consistent with the sign
of the gradient (Fig. 3).
For
sufficiently powerful sources, close to those used in
[5],
the destruction of the flow type
occurs
regardless of whether they are
placed in the negative or positive region of the gradient. Thus, the
nonlinearity of the problem does not allow using the maximum pressure
gradient
field
with respect to heat
sources to select control actions.
However, when the sources are placed
in a region with zero gradient, the flow type
will
remain the same for much more powerful sources.
Considered
the following ways to control the flow in the interaction
mode
using
distributed
heat sources:
·
Deflection
of an incident shock wave when passing through a heated region with
transformation of the interaction of shocks from type
to type
.
·
Formation
of a “thermal needle”
using
a local intense heat source.
·
-to-type
shock interactions
using distributed heat sources of moderate intensity
.
The calculation
results indicate that the transformation of interaction
type
into type
requires less energy. It is also possible to purposefully transform a
type structure
into a type
if information about the desired
flow field is available.
Solving the problem
of finding heat sources that reduce the pressure on the surface of the cylinder
during the interaction
of shocks is
possible using gradient optimization and solving conjugate equations. Numerical
experiments have shown that both the integrals
of the pressure functions on the
body surface and the discrepancy between the current and target flow parameters
in the computational field can be used as target functionals .
In this case, the search for the source with the minimum
integral value significantly depends on the choice of the regularization
coefficient. Visual representation of the flow control process is an integral
part of process control.
When solving the problem of controlling
the interaction of shocks, it is necessary to simultaneously visualize the flow
field, the field of conjugate parameters (sensitivity coefficients) and the
field of heat sources
,
which makes it possible to
determine the zones and intensity of the impact and the corresponding results.
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