The 2nd China Aeronautical Science & Technology Conference
Generation of Dynamic Grids and Computation of Unsteady
Transonic Flows around Assemblies
Cai Fei1,*, Zhang Liping2
1Department
2School
of Aerodynamics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
of Electronic and Information Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract
The coupled numerical simulation of flow field, solid temperature field, species concentration field and gas radiation transfer/
energy field based on statistical narrow-band correlated-k (SNBCK) model, is employed to accurately predict aerothermodynamic
characteristic of aircraft exhaust system. A series of methods to increase computational efficiency and descend computational
resources make it possible to finish the calculation in PC. The parameters of narrow-band model are evaluated by HITEMP
line-by-line database. Three examples have proved the accuracy of using these methods to solve flow heat transfer coupled problem and radiation transfer/energy equation, which are the calculation of temperature distribution of water-cooling nozzle in rocket engine, the calculation of carbon dioxide absorptivity at 4.3 micron band, and the gas radiation heat transfer evaluation of the
cylindrical furnace. Finally, the inner flaps temperature distribution of ejecting nozzle with floating outer flaps is computed, under high-altitude, high-speed and afterburning conditions. Two completely different air-inlet schemes of ejecting channel almost
achieve the same effect in cooling inner flaps.（英文摘要应在 150-200 词左右；包含研究的目的、方法、结果和结论；通常
用一般现在时，除非有明显的过去时间指示词出现 Abstract should be about 150 to 200 words, including purpose, method,
results and conclusion, present tense is preferred）
Keywords: transonic flow; unsteady flow; full-potential equation（5-8 个，建议其中 1-2 个选自 EI controlled term 词表 About
5-8 words. Selection of 1-2 from EI controlled term list is preferred）
1. Introduction1
The computation method of unsteady transonic flow
based on N-S equations should be best accurate, but to
three-dimensional complex problems, it can be
achieved only on large computers, and moreover, the
results are not ideal sometimes[1]. A viscous/inviscid
interaction method is an applicable one and the computation time can be reduced by two orders.
The method to solve the full-potential equation over
transonic unsteady forces on wings has been developed[2-3], but to assemblies, it is not mature, and the
research works are continuously carried on[4-5].
The key to compute the unsteady aerodynamic forces is to generate dynamic grids efficiently. In the present paper, an algebraic method and rapid deforming
technique are used to generate three-dimensional
boundary-fitted dynamic grids for assemblies. The
conservative full-potential equation is solved by a
time-accurate approximate factorization algorithm and
internal Newton iterations. An integral boundary layer
method based on the dissipation integral is used to ac*Corresponding author. E-mail address: abc@buaa.edu.cn
Foundation item: National Natural Science Foundation of China (基金
号)
ISBN-XXXX/$ - (留空由我会统一填写 for CSAA)
count for viscous effects. The computational results
about unsteady transonic forces on wings, bodies and
control surfaces are in agreement with experimental
data.（前言一般不要放图、
表、
公式 Equations, figures
and tables are usually not supposed to appear in this
part.）
2. Computation Scheme
2.1. Governing equation
The unsteady full-potential equation written in a
body fitted coordinate system is given by
(  J )  ( UJ )  ( VJ )  ( WJ )  0
(1)
where  is density, U, V, and W are the contravariant
velocity components in the , , and  directions,  means time, and J is Jacobian. （正文中不要
用公式编辑器插入符号，直接敲入即可 do not use
Equation Editor to insert symbols but type them directly）
q 2  (U  t )  (V  t )  (W   t )

U  t  a11  a12  a13

V  t  a12  a22  a23
W    a   a   a 
t
13 
23 
33 

where subscript s represents the boundary-fitted static
grids which are used as the initial values, r represents
the instantaneous coordinate that the static grids move
to with bodies, g is the function about the serial number of the grid lines.
（公式编辑器中的行距不要改动，默认值即可 Do
not change the line-width in the Equation Editor）
(公式中用到小括号时，不要用公式编辑器中的括
号，如  x  ，直接用键盘敲入即可如 ( x) When there
is parenthesis in an equation, do not use the parenthesis
in the Equation Editor but type it through keyboard.
 x  .)
For instance, use ( x) instead of
where is velocity potential.
2.2. Generation of grids
The algebraic method is used to generate
three-dimensional boundary-fitted grids for a wing.
The height of the first grid next to the body is controlled, and the grids near to the body are normalized.
To wing/missile combination, the H-H boundaryfitted grids for a single wing are generated at first in
order to simulate the aerodynamic forces on missiles
accurately. The section at the position of pylon is
changed into two overlapping layers. A partial deforming technique is used in the area of pylon/launching/
missile to get the static grids for the combination.
When with fuselage, the deforming technique is used
at the symmetric section to get the static grids also.
To simplify the equation, the dynamic grids where
the far field is fixed and the body is dynamic are applied to unsteady flows. As it will consume much time
to generate grids once a time-step, to aerodynamic
elasticity movement of small amplitude, ordinarily, the
grids are obtained by solving the balance equation
based on last time-step, but that is still not economic.
Taking the incompressible potential flow round a
cylinder for example, the stream function is

  V  r 

2
a
r

 sin 

(2)
where a is radius, V is the velocity of free stream.
Magnifying its radius to a   , then the previous
streamline passing point (r, ) will yield a normal displacement r
r     
 r  a 2
r 2  a2
(3)
That is difficulty directly to generate dynamic grids.
Simplifying further, the coordinates (represented by
subscript u) of the dynamic grids are
xu  xr  ( xr  xs )  g
(4)
  i  i 2  j  j 2  k  k 2 
b
b
b
g  max  
 ,
 ,
 
  if  ib   jf  jb   kf  kb  


(5)
where b represents the corresponding boundary points,
f represents the far field points.
3. Presentation of Result
F5 wing/missile and HARW wing/fuselage are taken
for examples in the present paper. Fig.1 shows the
sketches of F5 wing/missile and HARW wing/fuselage
with control surfaces. The missile is located at 73.5
percent of half-span. The trailing edge control surface
of HARW wing/fuselage takes up 20 percent of the
chord; control surface I is located at a position from
58.9% to 79.4% of half span; control surface II is located from 9.6% to 23.9%. Fig.2 is the H-H surface
grids for F5 wing/missile. Fig.3 shows the overlooking
view of HARW wing/fuselage assembles’ C-H grids.
Fig.4 Comparison between computation result and experimental data[24] for temperature distribution of inner
wall of nozzle.
Table 5 CPU time ratio of each term
Computational term
CPU time/%
Flow field
32.6
Solid temperature field
2.2
Species concentration field
4.3
Radiation transfer/energy field
60.9
4. Conclusions
(1) A rapid method of the generation of boundary-fitted dynamic grids is developed in this paper, and
the method of Viscous/Inviscid Interaction is used to
compute the unsteady aerodynamic forces on
wing/missiles and wing/body with control surfaces.
(2) The computation results are in agreement with
experimental data. To unsteady flows, it takes nearly
one hour to carry out three time cycles with 140 thousand nodes, so this method is powerful. This method
presented in this paper provides an applicable, efficient
tool for the aerodynamic elasticity analysis in the transonic aspect. （分点总结，只写结论，其他背景、方法
[5]
[6]
都不必赘述 only conclusions, no background information and methods）
[7]
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