We consider the following semi-linear equations
(
-
Δ
)
p
u
=
u
+
γ
in
R
n
,
where
γ
∈
(
1
,
n
+
2
p
n
-
2
p
)
,
n
>
2
p
>
0
,
u
+
=
max
{
u
,
0
}
, and
2
≤
p
∈
N
or
p
∈
(
0
,
1
)
. Subject to the ...integral constraint
u
+
γ
∈
L
1
(
R
n
)
,
we obtain the classification of solutions to the above polyharmonic equation for any
γ
<
n
+
2
p
n
-
2
p
and
γ
≤
n
n
-
2
p
, according to the two different assumptions:
Δ
u
(
x
)
→
0
and
u
(
x
)
=
o
(
|
x
|
2
)
at infinity, respectively. Under the other integral constraint
u
+
q
∈
L
1
(
R
n
)
,
q
=
n
(
γ
-
1
)
2
p
,
γ
<
n
+
2
p
n
-
2
p
,
which is scaling invariant, the classification of solutions with the decay assumption
Δ
u
(
x
)
→
0
at infinity is established for any integer
p
≥
2
, and the classification of solutions with the growth assumption
u
(
x
)
=
o
(
|
x
|
2
)
at infinity is proved for integers
p
=
2
,
3
as well. In the fractional equation case, namely
p
∈
(
0
,
1
)
, under either of the above two integral constraints, we also complete the classification of solutions with certain growth assumption at infinity.
In this paper, Computational Fluid Dynamics (CFD) simulations are performed to investigate the impingement cooling on internal leading edge region which is stretched by the middle cross section of ...the first stage rotor blade of GE-E
3 engine high pressure gas turbine. The simulations are carried out for a blade with a single row of circle jets at five different positions and seven different inlet flow Mach numbers. The results indicate that the global area weighted average Nusselt number at the blade leading edge increases with the increase of jet Mach number, and increases with the decrease of the distance between the jet nozzle and the pressure side. The correlation for the area weighted average Nusselt number as a function of the parameters is derived for the range of the parameters considered. The streamwise length weighted average Nusselt number and the spanwise length weighted average Nusselt number also increase with the decrease of the spacing between the jet nozzle and the pressure side, and increase with the increase of jet Mach number. The side entry jet is desirable to improve the performance of impingement cooling on turbine leading edge, but the arrangement of the jet nozzle and the shape of the internal cooling passage should be further optimized to improve the distribution of the heat transfer coefficient.
Impingement cooling is a widely used cooling technology with extremely high local heat flux dissipation, but its heat transfer is often destroyed by upstream crossflow. To improve the impingement ...heat transfer under crossflow condition, a skewed-twisted rib pair is proposed for simplified leading edge structure. The flow and heat transfer characteristics of Baseline and STRP cases are compared when the crossflow ratio varies from 0.2 to 0.4. According to the results, in both Baseline and STRP cases, increased crossflow ratio results in a decrease in impingement heat transfer, and it also leads to the peak value location of Nusselt number shift further downstream. Besides, with the increase of crossflow ratio, the total pressure-drop of jet decreases, but that of crossflow increases. Under the same crossflow ratio, the arrangement of STRP intensifies the turbulence level and induces an additional vortex pair whose centerline direction is same as jet, so the impingement heat transfer performance is improved effectively. Moreover, the effect of skewed-twisted rib pair height is considered, and it is found that the increased height makes the heat transfer enhancement effect get obvious, but it also leads to an increase in total pressure-drops of both jet and crossflow.
Mortise and tenon are very important parts of gas turbine dealing operation safety. Additionally, the temperature distribution of the turbine blade and disk is affected by the heat transfer ...characteristics in its gap. Then, the S-shaped mortise and tenon gap were numerically studied under rotating conditions, and the flow and heat transfer characteristics were analyzed. First, the heat transfer coefficient (HTC) of the mortise and tenon surfaces was measured with thermochromic liquid crystal. Then, the numerical method was verified using the test results, and the grid independence analysis was conducted. Finally, the flow and heat transfer characteristics of the gap under static and rotating conditions were numerically studied, five different Reynolds numbers (Re = 15,000, 20,000, 25,000, 30,000, 35,000) and five gap widths (d = 1 mm, 1.5 mm, 2 mm, 2.5 mm, 3 mm) were conducted and analyzed in detail. The results show that, under the rotating condition, the pressure distribution in the gap is different from that of the static condition; the pressure increases along the radial direction due to the action of centrifugal force and reaches its maximum value at the corner of the “S” shaped structure. With the increase in Re, the heat transfer intensity of the gap increases gradually. Additionally, the heat transfer intensity of the gap increases with an increase in its width.
The current carbon-based energy system is undergoing a profound change driven by the increased concerns over the longevity and security of energy supply, as well as energy-related emissions of carbon ...dioxide and air pollutions. The evolutionary trend of this transition is toward a smart energy network of the future that is characterized by widespread deployment of clean energy technologies and intelligent energy management technologies. In this transition, hydrogen and fuel/electrolysis cell technologies have crucial roles to play in developing the smart energy network, which is particularly captured in this work. The features of the future energy system, i.e., the smart energy network, are illustrated. In particular, the visions from technical aspects for the deployment of battery-based electric vehicles and fuel-cell-based electric vehicles are discussed. The key technologies and its current status for the smart energy network are reviewed.
•The features for a smart energy network of the future are identified.•The roles of fuel/electrolysis cells in the smart energy network are illustrated.•The visions for the deployment of EVs based on battery and fuel cell are discussed.•The future smart grid is an integration of electric grid, thermal grid and fuel grid.•The key technologies and its state-of-art for the smart energy network are reviewed.
Characteristics of particle migration and deposition were numerically investigated in presence of aggressive swirl at the turbine inlet. The isolated effects of the inlet swirl were considered in ...detail by shifting the circumferential position of the swirl and by implementing positive and negative swirling directions. Particles were released from the turbine inlet and the resulting deposition on the vanes was determined by using the critical velocity model in a range of particle diameters from 1 to 25 μm. Results show that the particles are more likely to move outwards to the boundary walls of the passage by the action of the swirling flow. However, this could be relieved by increasing the particle size. An imbalance problem of the deposition is found between the adjacent vanes, which could introduce additional inlet non-uniformities towards the downstream rotor and thus accelerate performance degradation of the turbine stage. Overall, the negative swirl case has higher overall capture efficiency within the entire turbine than the positive swirl case for larger particles, and when the inlet swirl is shifted to the mid-passage of the turbine, more deposits could be produced in comparison with the case in which the swirl aims at the vane leading edge.
During the practical operation of gas turbines, relatively cooled air from the compressor and the rim seal is applied in order to prevent mainstream ingestion into the space between the rotor and ...stator disc cavities, which can prolong the service life of hot components. On the one hand, the purge flow from the rim seal will inevitably interact with the mainstream and result in secondary flow on the endwall. On the other hand, it can also provide an additional cooling effect. In this paper, four rim seal structures, including an original single-tooth seal (ORI), a double-tooth seal (DS), a single-tooth seal with an adverse direction of the coolant purge flow and mainstream (AS) and a double-tooth seal with an adverse direction of the coolant purge flow and mainstream (ASDS), are experimentally and numerically investigated with mass flow ratios of 0.5%, 1.0% and 1.5%. The flow orientation of the coolant from the rim seal is considered as one of the main factors. The pressure-sensitive paint technique is used to experimentally measure the film cooling effectiveness on the endwall, and flow field analysis is conducted via numerical simulations. The results show that the cooling effect decreases in the cases of DS and ASDS. AS and ASDS can achieve a better film cooling performance, especially under a higher mass flow ratio. Furthermore, the structural changes in the rim seal have little impact on the aerodynamic performance. AS and ASDS can both achieve a better aerodynamic and film cooling performance.
In this paper numerical simulation is performed to simulate the impingement and film composite cooling on blade leading edge region. The relative performances of turbulence models are compared with ...available experimental data, and the results show that SST k–ω model is the best one based on simulation accuracy. Then the SST k–ω model is adopted for the simulation. The grid independence study is also carried out by using the Richardson extrapolation method. A single array of circle jets and three rows of film holes are arranged to investigate the impingement and film composite cooling. Five different blowing ratios and five different film hole spanwise angles are studied in detail. The results indicate that the heat transfer coefficient on the internal surface of turbine blade leading edge increases with the blowing ratio, and slightly changes with film hole spanwise angle. And the external film cooling effectiveness distribution would vary rapidly with the blowing ratio and the film hole spanwise angle.
•The performances of different turbulence model are validated.•Effects of blowing ratio and spanwise angle on composite cooling are investigated.•Spanwise angle is more effective to improve film cooling than impingement cooling.•Proper blowing ratio and lower spanwise angle is suitable for cooling design.
We consider periodic solutions of the following problem associated with the fractional Laplacian:
in
. The smooth function
is periodic about
and is a double-well potential with respect to
with wells ...at
and -1 for any
. We prove the existence of periodic solutions whose periods are large integer multiples of the period of
about the variable
by using variational methods. An estimate of the energy functional, Hamiltonian identity and Modica-type inequality for periodic solutions are also established.
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