# Numerical Calculation of Effect of Elastic Deformation on Aerodynamic Characteristics of a Rocket

Authors: Laith K. Abbas; Dongyang Chen; Xiaoting Rui
Journal: International Journal of Aerospace Engineering
 (2014)

---

## Abstract

The application and workflow of Computational Fluid Dynamics (CFD)/Computational Structure Dynamics (CSD) on solving the static aeroelastic problem of a slender rocket are introduced. To predict static aeroelastic behavior accurately, two-way coupling and inertia relief methods are used to calculate the static deformations and aerodynamic characteristics of the deformed rocket. The aerodynamic coefficients of rigid rocket are computed firstly and compared with the experimental data, which verified the accuracy of CFD output. The results of the analysis for elastic rocket in the nonspinning and spinning states are compared with the rigid ones. The results highlight that the rocket deformation aspects are decided by the normal force distribution along the rocket length. Rocket deformation becomes larger with increasing the flight angle of attack. Drag and lift force coefficients decrease and pitching moment coefficients increase due to rocket deformations, center of pressure location forwards, and stability of the rockets decreases. Accordingly, the flight trajectory may be affected by the change of these aerodynamic coefficients and stability.


---

## Body

## 1. Introduction

Aeroelastic is a discipline which is used to study aerodynamics and elastic structure interaction and its application. A slender rocket is usually to be faced with serious aeroelastic problems when light weight and low stiffness structures are used. Aeroelastic problems should be considered in the early phase of the slender rocket structural design since any unstable response to aerodynamic loading may quickly lead to catastrophic structural failure, which may need usually expensive modifications. Flight tests and wind tunnels are two expensive ways performed in the late phase of the design. Therefore, computational aeroelasticity dynamics methods are used in order to calculate aeroelastic characteristics of the rocket during its development stages. With Computational Fluid Dynamics (CFD) technology and the increasing advance of computing power, computer aided engineering (CAE) analysis which integrated CFD analysis accurately computes the aerodynamic coefficients of many complex geometry [1] and complex flow phenomena [2, 3] and Computational Structure Dynamics (CSD) which solves the structural static and dynamic problems [4, 5] is a useful method to solve aeroelastic problems of flight vehicles accurately [6–8].In this paper, in order to solve the static aeroelastic problem of a slender wrap fins rocket in the linear angle-of-attack range, a two-way coupling approach is developed. To conduct this static aeroelastic analysis, ANSYS Workbench multiphysics coupling platform, which includes fluid, structural solvers, and coupling module, namely, FLUENT, ANSYS Mechanical, and System coupling, is used to solve the displacements associated with the aerodynamic pressure loading and find out the static aeroelastic behavior of this rocket in the nonspinning and spinning state. The single reference frame (SRF) model is used to perform the problem of spinning rocket [9]. CFD grid is deformed to be able to obtain aerodynamic solutions of the deformed geometry using dynamic mesh which is called diffusion-based smoothing method. For the structural analysis, the ANSYS Mechanical inertia relief option [10, 11], which is based on d’Alembert principle and used to simulate unconstrained rockets in flight and make sure the rocket has no rigid body displacement, is used with the linear elastic solver. The aerodynamic coefficients distribution of the spinning rigid rocket and the elastic one are calculated and compared with that of nonspinning ones. The steps of static aeroelastic calculations are described and the coupling procedure is explained in the following section.


## 2. Static Aeroelastic Calculations

### 2.1. Problem Definition

In the reality, the rigid geometry assumption holds for many engineering problems. In many cases, where the structures are flexible, fluid-structure interactions become important. In this work, a slender wrap fins rocket is studied. Static aeroelasticity considers the nonoscillatory effects of aerodynamic forces acting on the elastic structure [12]. Because of the elastic nature of the slender rocket, aerodynamic forces acting on the rocket contribute to structural deformation. This deflection of the structure tends to redistribute the aerodynamic forces acting on the rocket, and this interaction continues by leading to each other. For accurately simulating the static aeroelastic of free flight rocket, spinning is considered and structured mesh is adopted. As a result, a coupling approach is developed to solve the static aeroelastic problem as a significant part of rocket design workflow.


### 2.2. Solution Procedure

Two-way fluid structure method flow chart is shown in Figure1.Figure 1
Scheme of CAE code ANSYS Workbench.The static aeroelastic solutions of the nonspinning and spinning rockets are obtained using the following steps.(1)
Generate rocket geometry and fluid control volume by SPACECLAIM.
(2)
Generate a structured mesh by ICEM-CFD around the rigid rocket geometry.
(3)
Obtain a steady-state CFD solution for the rigid rocket by FLUENT.
(4)
Export the pressures on the rocket surface in ANSYS Mechanical by System coupling module.
(5)
Map the pressure at the CFD grid points to forces on the CSD nodes.
(6)
Obtain the structural response of the rocket by CSD code of ANSYS Mechanical.
(7)
Map the displacement at the CSD nodes to the displacements on the CFD grid points of the rocket surface by System coupling module.
(8)
Deform the whole CFD mesh.
(9)
Obtain steady-state CFD solution for the deformed rocket by FLUENT.
(10)
Repeat steps 4–9 until the displacement and aerodynamic coefficients do not change.


### 2.3. Numerical Method

#### 2.3.1. Geometry Model and Initial Conditions

The model used in this work is a wrap fins rocket with a slenderness ratio of more than 25. For comparing with the aerodynamic loads distribution of the elastic slender rocket, it is divided into many segments along the rocket body. Then, through computation, the aerodynamic parameters of every segment are obtained. The slender rocket consists of three parts as shown in Figure2(a). Longitudinally, the body is divided into 31 segments as tabulated in Table 1. The middle point of each section is the normal force action point. The geometry model of the rocket and the flow control volume (see Figure 2(b)) are established. The far-field should be placed far enough from the rocket since the free stream of infinity is defined as boundary conditions. For supersonic computation, the rocket has the far-field boundary located at about 5
D (where D is the rocket diameter) upstream away from the rocket-nose, about 30
D away from the rocket-bottom downstream, and about 15
D away from the rocket-surface in a lateral (see Figure 2(b)).Table 1
Rocket segmentation.
Part 1 (x
1
/
D)
1
2
2.3
Part 2 (x
2
/
D)
0.2
0.4
0.6
0.8
1
1.5
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Part 3 (x
3
/
D)
2Model geometry and flow control volume: (a) longitudinal division of the slender rocket and (b) flow control volume.
(a)
(b)The aerodynamic parameters distribution of the rigid rocket are calculated in the conditions that the angle of attacks is 2° and 4°, while the Mach numbers are 1.5, 2, 2.5, and 3, respectively. The rocket three parts maintain the defined density, Poisson’s ratio, and Young’s modulus, respectively (see Table2).Table 2
Material properties for rocket.
Part name
Density
Young’s modulus
Poisson’s ratio
Part 1, Part 3
7850
2
E
+
11
0.3
Part 2
2770
7.1
E
+
10
0.33


#### 2.3.2. CFD Governing Equations

For CFD problems, Direct Numerical Simulation (DNS) solves the Navier-Stokes (N-S) equations directly without the turbulent model needing high speed and large memory computer due to their high nonlinearity and complexity. It is impossible to adopt this method in practical engineering. Decomposing the N-S equations into the RANS equations makes it possible to simulate engineering fluid dynamic problems.Reynolds averaged N-S equations (RANS) can be expressed as(1)
∂
Q
∂
t
+
∂
E
∂
x
+
∂
F
∂
y
+
∂
G
∂
z
=
∂
E
v
∂
x
+
∂
F
v
∂
y
+
∂
G
v
∂
z
,
where
(2)
Q
=
[
ρ
,
ρ
u
,
ρ
v
,
ρ
w
,
ρ
E
]
T
,
E
=
[
ρ
u
,
ρ
u
2
+
p
,
ρ
u
v
,
ρ
u
w
,
(
ρ
E
+
p
)
u
]
T
,
F
=
[
ρ
v
,
ρ
v
u
,
ρ
v
2
+
p
,
ρ
v
w
,
(
ρ
E
+
p
)
v
]
T
,
G
=
[
ρ
w
,
ρ
w
u
,
ρ
w
v
,
ρ
w
2
+
p
,
(
ρ
E
+
p
)
w
]
T
,
E
v
=
[
0
,
τ
x
x
,
τ
x
y
,
τ
x
z
,
β
x
]
T
,
F
v
=
[
0
,
τ
y
x
,
τ
y
y
,
τ
y
z
,
β
y
]
T
,
G
v
=
[
0
,
τ
z
x
,
τ
z
y
,
τ
z
z
,
β
z
]
T
,
where ρ is the fluid density, (
u
,
v
,
w
) are the three components of the Cartesian coordinate system, p is pressure, and E is total energy of unit mass.N-S equations of rotating coordinates [9] are
(3)
∂
ρ
∂
t
+
∂
∂
x
j
(
ρ
w
j
)
=
0
,
∂
∂
t
(
ρ
u
i
)
+
∂
∂
x
j
(
ρ
w
j
w
i
+
p
δ
i
j
)
=
∂
∂
x
j
(
τ
i
j
)
+
ρ
F
i
,
∂
∂
t
(
ρ
E
′
)
+
∂
∂
x
j
(
ρ
w
j
E
′
+
w
j
p
)
=
∂
∂
x
j
(
w
i
τ
i
j
+
k
∂
T
∂
x
j
)
+
ρ
q
-
+
ρ
f
i
w
i
,
where
(4)
τ
i
j
=
μ
[
(
∂
w
i
∂
x
j
+
∂
w
j
∂
x
i
)
-
2
3
δ
i
j
∂
w
k
∂
x
k
]
,
F
i
=
f
i
-
2
e
i
j
k
ω
j
w
k
+
∂
∂
x
j
(
ω
2
R
2
2
)
,
E
′
=
e
+
W
2
2
-
ω
2
R
2
2
=
P
ρ
(
γ
-
1
)
+
W
2
2
-
ω
2
R
2
2
,
where W is the relative speed, E
′ is the total energy of relative rotation, and q
- is the radiant heat (q
-
=
0 in the present study). F is the external force, including volume force and centrifugal force. ω is the angular velocity of rotation and R is the radius of gyration.For the closure of the above equations, the shear stress transport (SST)k
-
ω turbulence model developed by Menter [13, 14] k
-
ε and standard k
-
ω model boundary layer inside and outside. k and ω transport equation can be written as [14]
(5)
d
(
ρ
k
)
d
t
=
τ
i
j
∂
u
i
∂
x
j
-
β
*
ρ
ω
k
+
∂
∂
x
j
[
(
μ
+
σ
k
μ
t
)
∂
k
∂
x
j
]
,
d
(
ρ
ω
)
d
t
=
γ
ρ
μ
t
τ
i
j
∂
u
i
∂
x
j
-
β
ρ
ω
2
+
∂
∂
x
j
[
(
μ
+
σ
ω
μ
t
)
∂
ω
∂
x
j
]
+
2
ρ
(
1
-
F
1
)
σ
ω
2
1
ω
∂
k
∂
x
j
∂
ω
∂
x
j
,
where τ
i
j is the shear stress:
(6)
τ
i
j
=
μ
t
(
∂
u
i
∂
x
j
+
∂
u
j
∂
x
i
-
2
3
∂
u
k
∂
x
k
δ
i
j
)
-
2
3
ρ
k
δ
i
j
.
Blending function F
1is
(7)
F
1
=
tanh
(
ar
g
1
4
)
,
where
(8)
ar
g
1
=
min
⁡
[
max
⁡
(
k
0.09
ω
y
,
500
υ
y
2
ω
)
,
4
ρ
σ
ω
2
k
C
D
k
ω
y
2
]
,
C
D
k
ω
=
max
⁡
(
2
ρ
σ
ω
2
1
ω
∂
k
∂
x
j
∂
ω
∂
x
j
,
1
0
-
20
)
.
The eddy viscosity is defined as
(9)
μ
t
=
ρ
α
1
k
max
⁡
(
α
1
ω
,
Ω
F
2
)
,
where Ω is the absolute value of the vorticity.F
2 is given by
(10)
F
2
=
tanh
(
ar
g
2
2
)
,
where ar
g
2
=
max
⁡
(
2
k
/
0.09
ω
y
,
500
μ
/
ρ
y
2
ω
).The constantsϕ of the SST model are calculated from the constants, ϕ
1
,
ϕ
2 as follows:
(11)
ϕ
=
F
1
ϕ
1
+
(
1
-
F
1
)
ϕ
2
,
where set 1
(
ϕ
1
) is the constants of k
-
ω model and set 2
(
ϕ
2
) is the constants of k
-
ε model.The constants of set1
(
ϕ
1
) are
(12)
σ
k
1
=
0.5
,
σ
ω
1
=
0.5
,
β
1
=
0.075
,
β
*
=
0.09
,
κ
=
0.41
,
γ
1
=
(
β
1
β
*
)
-
(
σ
ω
1
κ
2
β
*
)
.
The constants of set 2
(
ϕ
2
) are
(13)
σ
k
2
=
1.0
,
σ
ω
1
=
0.856
,
β
2
=
0.0828
,
β
*
=
0.09
,
κ
=
0.41
,
γ
2
=
(
β
2
β
*
)
-
(
σ
ω
2
κ
2
β
*
)
.
All other parameters are given in [14].In this work, the far-field free stream condition is standard temperature and pressure (101.325 kPa, 288 K) and the far-field boundary is based on Riemann invariants reflecting boundary conditions. The single reference frame (SRF) model is used to perform the problem of spinning rocket. The air is assumed as an ideal gas and the viscosity varies with temperature in accordance with Sutherland three coefficient formulas. The solution method is implicit formulation and Advection Upstream Splitting Method (AUSM). AUSM which is based on the finite volume method and is used to Spatial discrete the space of fluid field. Second Order Upwind scheme is used on convection and turbulent viscosity terms. Flow field mesh number is 800000,Y
+
=
30
-
100 [15]. Flow field computation grid is shown in Figures 3(a) and 3(b).Discrete mesh: (a) flow field grids and (b) the section of grids.
(a)
(b)Solving the above equations, density, velocity, pressure, and so forth could be computed at every grid node. Then, normal, axial, lift, drag, pitching moment, and center of the pressure coefficients can be obtained by following equations, respectively:(14)
C
N
=
N
q
∞
S
,
(15)
C
A
=
A
q
∞
S
,
(16)
C
L
=
C
N
×
cos
⁡
α
-
C
A
×
sin
α
,
(17)
C
D
=
C
N
×
sin
α
+
C
A
×
cos
⁡
α
,
(18)
C
m
=
pitching
moment
q
∞
S
l
,
(19)
x
c
p
=
x
c
g
-
(
C
m
C
N
)
,
where q
∞, α, S, x
c
p, and x
c
g are the dynamic pressure, angle of attack, reference area, center of pressure location, and center of gravity location, respectively. l is length of the rocket. The vertex of the warhead is the reference point.


#### 2.3.3. Static Structure Analysis Equations

A static structural analysis determines the displacements, stresses, strains, and forces in the structures or components caused by loads that do not induce significant inertia and damping effects. Steady loading and response conditions are assumed; that is, the loads and the structure’s response are assumed to vary slowly with respect to time. This is an isotropic linear elastic problem. The static structure analysis equation can be written as(20)
[
K
]
{
δ
}
=
{
F
}
,
where [
K
] is stiffness matrix, {δ} is displacement vector, and {
F
} is force vector.To acquire the high quality mesh of the rocket, 10 nodes tetrahedron element (SOLID187) which is used for the wrap fins and warhead and 20 nodes hexahedron element (SOLID186) which is used for simple geometric part are used to perform the structural analysis. For these linear elastic structural mechanics problems,K is a constant and aerodynamic force F is calculated by CFD code. Then, the deformation δ can be obtained. The rocket solid region grid is shown in Figure 4.Figure 4
Solid region grid.Multipoint Constraint (MPC) approach is used to define the bonding of contact surfaces. Normal force and tangential force of each part are transferred through contact surfaces. Inertia relief method, which is based on d’Alembert principle and used to simulate unconstrained rockets in flight and make sure the rocket has no rigid body displacement, is used with the linear solver. In addition, because the body is rotating, a rotating angular velocity is applied to the rocket body.Inertia relief is an approach in which the applied forces and torques are balanced by inertial forces induced by an acceleration field. Consider the application of an acceleration field (to be determined) that precisely balances the applied loads:(21)
{
F
t
a
}
+
[
M
t
]
{
a
t
1
}
=
{
0
}
,
{
F
r
a
}
+
[
M
r
]
{
a
r
1
}
=
{
0
}
,
where{
F
t
a
} = force components of the applied load vector,
{
F
r
a
} = moment components of the applied load vector,
{
a
t
1
} = translational acceleration vector due to inertia relief (to be determined),
{
a
r
1
} = rotational acceleration vector due to inertia relief (to be determined),
{
r
} = position vector = [
x
,
y
,
z
]
T,
[
M
t
] = mass tensor for the entire finite element model (developed below),
[
M
r
] = mass moments and mass products of the inertia tensor for the entire finite element model (developed below).Once [
M
t
] and [
M
r
] are developed and {
F
t
a
} and {
F
r
a
} can be obtained from CFD output, then {
a
t
1
} and {
a
r
1
} in (21) can be solved. The output inertia relief summary includes {
a
t
1
} (output as translational accelerations) and {
a
r
1
} (output as rotational accelerations).


#### 2.3.4. Coupling Boundary Conditions and Dynamic Mesh

In this fluid-structure interaction work, there is a boundary where the fluid and structural domains interactions occur at this boundary or interface. Deformation compatibility conditions and force equilibrium conditions should be satisfied on the fluid-structure coupling interface as [16]
(22)
d
f
=
d
s
,
n
·
τ
f
=
n
·
τ
s
,
q
f
=
q
s
,
T
f
=
T
s
,
where d, q, T, and τ are displacement, heat flux, temperature, and stress field on the fluid-structure coupling interface, respectively. n is the normal direction of interface. Subscripts fand s represent the fluid and the solid, respectively.For data transfer of the dissimilar meshes, the interpolation operation is an essential step. The interpolation called globally conservative is used in this paper. It collects the interface meshes from the CFD code, does the mapping, and communicates stagger loop controls to the CFD code. The mapping generated by ANSYS code is used to interpolate loads between dissimilar meshes on either side of the coupling interface. Each field solver advances through a sequence of multifield stagger (coupling) iterations within each coupling step. During every stagger iteration, each field solver collects the loads that it requires from the other field solvers and then solves its physics fields.The flow field grids motion is achieved by diffusion-based smoothing method which is based on cell volume. And the mesh motion is governed by the diffusion equation:(23)
∇
·
(
γ
∇
u
¯
)
=
0
,
where u
¯ is the mesh displacement velocity and γ is the diffusion coefficient
(24)
γ
=
1
V
α
,
where V is the normalized cell volume and α is the user input parameter (α
=
1.9 in the present study). The boundary conditions for (23) are obtained from the computed boundary motion.Equation (23) is discretized by finite volume method, and the resulting matrix is solved iteratively using the Algebraic Multigrid (AMG) solver. The cell centered solution for the displacement velocity u
¯ from (23) is interpolated onto the nodes using inverse distance weighted averaging, and the node positions are updated according to [17]:
(25)
x
¯
new
=
x
¯
old
+
u
¯
Δ
t
.For (25), Δ
t is one coupling step that includes 1~3 iterations and one iteration includes 30 fluid solver iteration steps which make sure every coupling iteration step is converged. Use this method, decreasing the diffusivity in larger cells causes those cells to absorb more of mesh motion. It is better to preserve the cell quality of smaller cells which are close to the boundary motion and therefore preserve the computational accuracy.


## 2.1. Problem Definition

In the reality, the rigid geometry assumption holds for many engineering problems. In many cases, where the structures are flexible, fluid-structure interactions become important. In this work, a slender wrap fins rocket is studied. Static aeroelasticity considers the nonoscillatory effects of aerodynamic forces acting on the elastic structure [12]. Because of the elastic nature of the slender rocket, aerodynamic forces acting on the rocket contribute to structural deformation. This deflection of the structure tends to redistribute the aerodynamic forces acting on the rocket, and this interaction continues by leading to each other. For accurately simulating the static aeroelastic of free flight rocket, spinning is considered and structured mesh is adopted. As a result, a coupling approach is developed to solve the static aeroelastic problem as a significant part of rocket design workflow.


## 2.2. Solution Procedure

Two-way fluid structure method flow chart is shown in Figure1.Figure 1
Scheme of CAE code ANSYS Workbench.The static aeroelastic solutions of the nonspinning and spinning rockets are obtained using the following steps.(1)
Generate rocket geometry and fluid control volume by SPACECLAIM.
(2)
Generate a structured mesh by ICEM-CFD around the rigid rocket geometry.
(3)
Obtain a steady-state CFD solution for the rigid rocket by FLUENT.
(4)
Export the pressures on the rocket surface in ANSYS Mechanical by System coupling module.
(5)
Map the pressure at the CFD grid points to forces on the CSD nodes.
(6)
Obtain the structural response of the rocket by CSD code of ANSYS Mechanical.
(7)
Map the displacement at the CSD nodes to the displacements on the CFD grid points of the rocket surface by System coupling module.
(8)
Deform the whole CFD mesh.
(9)
Obtain steady-state CFD solution for the deformed rocket by FLUENT.
(10)
Repeat steps 4–9 until the displacement and aerodynamic coefficients do not change.


## 2.3. Numerical Method

### 2.3.1. Geometry Model and Initial Conditions

The model used in this work is a wrap fins rocket with a slenderness ratio of more than 25. For comparing with the aerodynamic loads distribution of the elastic slender rocket, it is divided into many segments along the rocket body. Then, through computation, the aerodynamic parameters of every segment are obtained. The slender rocket consists of three parts as shown in Figure2(a). Longitudinally, the body is divided into 31 segments as tabulated in Table 1. The middle point of each section is the normal force action point. The geometry model of the rocket and the flow control volume (see Figure 2(b)) are established. The far-field should be placed far enough from the rocket since the free stream of infinity is defined as boundary conditions. For supersonic computation, the rocket has the far-field boundary located at about 5
D (where D is the rocket diameter) upstream away from the rocket-nose, about 30
D away from the rocket-bottom downstream, and about 15
D away from the rocket-surface in a lateral (see Figure 2(b)).Table 1
Rocket segmentation.
Part 1 (x
1
/
D)
1
2
2.3
Part 2 (x
2
/
D)
0.2
0.4
0.6
0.8
1
1.5
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Part 3 (x
3
/
D)
2Model geometry and flow control volume: (a) longitudinal division of the slender rocket and (b) flow control volume.
(a)
(b)The aerodynamic parameters distribution of the rigid rocket are calculated in the conditions that the angle of attacks is 2° and 4°, while the Mach numbers are 1.5, 2, 2.5, and 3, respectively. The rocket three parts maintain the defined density, Poisson’s ratio, and Young’s modulus, respectively (see Table2).Table 2
Material properties for rocket.
Part name
Density
Young’s modulus
Poisson’s ratio
Part 1, Part 3
7850
2
E
+
11
0.3
Part 2
2770
7.1
E
+
10
0.33


### 2.3.2. CFD Governing Equations

For CFD problems, Direct Numerical Simulation (DNS) solves the Navier-Stokes (N-S) equations directly without the turbulent model needing high speed and large memory computer due to their high nonlinearity and complexity. It is impossible to adopt this method in practical engineering. Decomposing the N-S equations into the RANS equations makes it possible to simulate engineering fluid dynamic problems.Reynolds averaged N-S equations (RANS) can be expressed as(1)
∂
Q
∂
t
+
∂
E
∂
x
+
∂
F
∂
y
+
∂
G
∂
z
=
∂
E
v
∂
x
+
∂
F
v
∂
y
+
∂
G
v
∂
z
,
where
(2)
Q
=
[
ρ
,
ρ
u
,
ρ
v
,
ρ
w
,
ρ
E
]
T
,
E
=
[
ρ
u
,
ρ
u
2
+
p
,
ρ
u
v
,
ρ
u
w
,
(
ρ
E
+
p
)
u
]
T
,
F
=
[
ρ
v
,
ρ
v
u
,
ρ
v
2
+
p
,
ρ
v
w
,
(
ρ
E
+
p
)
v
]
T
,
G
=
[
ρ
w
,
ρ
w
u
,
ρ
w
v
,
ρ
w
2
+
p
,
(
ρ
E
+
p
)
w
]
T
,
E
v
=
[
0
,
τ
x
x
,
τ
x
y
,
τ
x
z
,
β
x
]
T
,
F
v
=
[
0
,
τ
y
x
,
τ
y
y
,
τ
y
z
,
β
y
]
T
,
G
v
=
[
0
,
τ
z
x
,
τ
z
y
,
τ
z
z
,
β
z
]
T
,
where ρ is the fluid density, (
u
,
v
,
w
) are the three components of the Cartesian coordinate system, p is pressure, and E is total energy of unit mass.N-S equations of rotating coordinates [9] are
(3)
∂
ρ
∂
t
+
∂
∂
x
j
(
ρ
w
j
)
=
0
,
∂
∂
t
(
ρ
u
i
)
+
∂
∂
x
j
(
ρ
w
j
w
i
+
p
δ
i
j
)
=
∂
∂
x
j
(
τ
i
j
)
+
ρ
F
i
,
∂
∂
t
(
ρ
E
′
)
+
∂
∂
x
j
(
ρ
w
j
E
′
+
w
j
p
)
=
∂
∂
x
j
(
w
i
τ
i
j
+
k
∂
T
∂
x
j
)
+
ρ
q
-
+
ρ
f
i
w
i
,
where
(4)
τ
i
j
=
μ
[
(
∂
w
i
∂
x
j
+
∂
w
j
∂
x
i
)
-
2
3
δ
i
j
∂
w
k
∂
x
k
]
,
F
i
=
f
i
-
2
e
i
j
k
ω
j
w
k
+
∂
∂
x
j
(
ω
2
R
2
2
)
,
E
′
=
e
+
W
2
2
-
ω
2
R
2
2
=
P
ρ
(
γ
-
1
)
+
W
2
2
-
ω
2
R
2
2
,
where W is the relative speed, E
′ is the total energy of relative rotation, and q
- is the radiant heat (q
-
=
0 in the present study). F is the external force, including volume force and centrifugal force. ω is the angular velocity of rotation and R is the radius of gyration.For the closure of the above equations, the shear stress transport (SST)k
-
ω turbulence model developed by Menter [13, 14] k
-
ε and standard k
-
ω model boundary layer inside and outside. k and ω transport equation can be written as [14]
(5)
d
(
ρ
k
)
d
t
=
τ
i
j
∂
u
i
∂
x
j
-
β
*
ρ
ω
k
+
∂
∂
x
j
[
(
μ
+
σ
k
μ
t
)
∂
k
∂
x
j
]
,
d
(
ρ
ω
)
d
t
=
γ
ρ
μ
t
τ
i
j
∂
u
i
∂
x
j
-
β
ρ
ω
2
+
∂
∂
x
j
[
(
μ
+
σ
ω
μ
t
)
∂
ω
∂
x
j
]
+
2
ρ
(
1
-
F
1
)
σ
ω
2
1
ω
∂
k
∂
x
j
∂
ω
∂
x
j
,
where τ
i
j is the shear stress:
(6)
τ
i
j
=
μ
t
(
∂
u
i
∂
x
j
+
∂
u
j
∂
x
i
-
2
3
∂
u
k
∂
x
k
δ
i
j
)
-
2
3
ρ
k
δ
i
j
.
Blending function F
1is
(7)
F
1
=
tanh
(
ar
g
1
4
)
,
where
(8)
ar
g
1
=
min
⁡
[
max
⁡
(
k
0.09
ω
y
,
500
υ
y
2
ω
)
,
4
ρ
σ
ω
2
k
C
D
k
ω
y
2
]
,
C
D
k
ω
=
max
⁡
(
2
ρ
σ
ω
2
1
ω
∂
k
∂
x
j
∂
ω
∂
x
j
,
1
0
-
20
)
.
The eddy viscosity is defined as
(9)
μ
t
=
ρ
α
1
k
max
⁡
(
α
1
ω
,
Ω
F
2
)
,
where Ω is the absolute value of the vorticity.F
2 is given by
(10)
F
2
=
tanh
(
ar
g
2
2
)
,
where ar
g
2
=
max
⁡
(
2
k
/
0.09
ω
y
,
500
μ
/
ρ
y
2
ω
).The constantsϕ of the SST model are calculated from the constants, ϕ
1
,
ϕ
2 as follows:
(11)
ϕ
=
F
1
ϕ
1
+
(
1
-
F
1
)
ϕ
2
,
where set 1
(
ϕ
1
) is the constants of k
-
ω model and set 2
(
ϕ
2
) is the constants of k
-
ε model.The constants of set1
(
ϕ
1
) are
(12)
σ
k
1
=
0.5
,
σ
ω
1
=
0.5
,
β
1
=
0.075
,
β
*
=
0.09
,
κ
=
0.41
,
γ
1
=
(
β
1
β
*
)
-
(
σ
ω
1
κ
2
β
*
)
.
The constants of set 2
(
ϕ
2
) are
(13)
σ
k
2
=
1.0
,
σ
ω
1
=
0.856
,
β
2
=
0.0828
,
β
*
=
0.09
,
κ
=
0.41
,
γ
2
=
(
β
2
β
*
)
-
(
σ
ω
2
κ
2
β
*
)
.
All other parameters are given in [14].In this work, the far-field free stream condition is standard temperature and pressure (101.325 kPa, 288 K) and the far-field boundary is based on Riemann invariants reflecting boundary conditions. The single reference frame (SRF) model is used to perform the problem of spinning rocket. The air is assumed as an ideal gas and the viscosity varies with temperature in accordance with Sutherland three coefficient formulas. The solution method is implicit formulation and Advection Upstream Splitting Method (AUSM). AUSM which is based on the finite volume method and is used to Spatial discrete the space of fluid field. Second Order Upwind scheme is used on convection and turbulent viscosity terms. Flow field mesh number is 800000,Y
+
=
30
-
100 [15]. Flow field computation grid is shown in Figures 3(a) and 3(b).Discrete mesh: (a) flow field grids and (b) the section of grids.
(a)
(b)Solving the above equations, density, velocity, pressure, and so forth could be computed at every grid node. Then, normal, axial, lift, drag, pitching moment, and center of the pressure coefficients can be obtained by following equations, respectively:(14)
C
N
=
N
q
∞
S
,
(15)
C
A
=
A
q
∞
S
,
(16)
C
L
=
C
N
×
cos
⁡
α
-
C
A
×
sin
α
,
(17)
C
D
=
C
N
×
sin
α
+
C
A
×
cos
⁡
α
,
(18)
C
m
=
pitching
moment
q
∞
S
l
,
(19)
x
c
p
=
x
c
g
-
(
C
m
C
N
)
,
where q
∞, α, S, x
c
p, and x
c
g are the dynamic pressure, angle of attack, reference area, center of pressure location, and center of gravity location, respectively. l is length of the rocket. The vertex of the warhead is the reference point.


### 2.3.3. Static Structure Analysis Equations

A static structural analysis determines the displacements, stresses, strains, and forces in the structures or components caused by loads that do not induce significant inertia and damping effects. Steady loading and response conditions are assumed; that is, the loads and the structure’s response are assumed to vary slowly with respect to time. This is an isotropic linear elastic problem. The static structure analysis equation can be written as(20)
[
K
]
{
δ
}
=
{
F
}
,
where [
K
] is stiffness matrix, {δ} is displacement vector, and {
F
} is force vector.To acquire the high quality mesh of the rocket, 10 nodes tetrahedron element (SOLID187) which is used for the wrap fins and warhead and 20 nodes hexahedron element (SOLID186) which is used for simple geometric part are used to perform the structural analysis. For these linear elastic structural mechanics problems,K is a constant and aerodynamic force F is calculated by CFD code. Then, the deformation δ can be obtained. The rocket solid region grid is shown in Figure 4.Figure 4
Solid region grid.Multipoint Constraint (MPC) approach is used to define the bonding of contact surfaces. Normal force and tangential force of each part are transferred through contact surfaces. Inertia relief method, which is based on d’Alembert principle and used to simulate unconstrained rockets in flight and make sure the rocket has no rigid body displacement, is used with the linear solver. In addition, because the body is rotating, a rotating angular velocity is applied to the rocket body.Inertia relief is an approach in which the applied forces and torques are balanced by inertial forces induced by an acceleration field. Consider the application of an acceleration field (to be determined) that precisely balances the applied loads:(21)
{
F
t
a
}
+
[
M
t
]
{
a
t
1
}
=
{
0
}
,
{
F
r
a
}
+
[
M
r
]
{
a
r
1
}
=
{
0
}
,
where{
F
t
a
} = force components of the applied load vector,
{
F
r
a
} = moment components of the applied load vector,
{
a
t
1
} = translational acceleration vector due to inertia relief (to be determined),
{
a
r
1
} = rotational acceleration vector due to inertia relief (to be determined),
{
r
} = position vector = [
x
,
y
,
z
]
T,
[
M
t
] = mass tensor for the entire finite element model (developed below),
[
M
r
] = mass moments and mass products of the inertia tensor for the entire finite element model (developed below).Once [
M
t
] and [
M
r
] are developed and {
F
t
a
} and {
F
r
a
} can be obtained from CFD output, then {
a
t
1
} and {
a
r
1
} in (21) can be solved. The output inertia relief summary includes {
a
t
1
} (output as translational accelerations) and {
a
r
1
} (output as rotational accelerations).


### 2.3.4. Coupling Boundary Conditions and Dynamic Mesh

In this fluid-structure interaction work, there is a boundary where the fluid and structural domains interactions occur at this boundary or interface. Deformation compatibility conditions and force equilibrium conditions should be satisfied on the fluid-structure coupling interface as [16]
(22)
d
f
=
d
s
,
n
·
τ
f
=
n
·
τ
s
,
q
f
=
q
s
,
T
f
=
T
s
,
where d, q, T, and τ are displacement, heat flux, temperature, and stress field on the fluid-structure coupling interface, respectively. n is the normal direction of interface. Subscripts fand s represent the fluid and the solid, respectively.For data transfer of the dissimilar meshes, the interpolation operation is an essential step. The interpolation called globally conservative is used in this paper. It collects the interface meshes from the CFD code, does the mapping, and communicates stagger loop controls to the CFD code. The mapping generated by ANSYS code is used to interpolate loads between dissimilar meshes on either side of the coupling interface. Each field solver advances through a sequence of multifield stagger (coupling) iterations within each coupling step. During every stagger iteration, each field solver collects the loads that it requires from the other field solvers and then solves its physics fields.The flow field grids motion is achieved by diffusion-based smoothing method which is based on cell volume. And the mesh motion is governed by the diffusion equation:(23)
∇
·
(
γ
∇
u
¯
)
=
0
,
where u
¯ is the mesh displacement velocity and γ is the diffusion coefficient
(24)
γ
=
1
V
α
,
where V is the normalized cell volume and α is the user input parameter (α
=
1.9 in the present study). The boundary conditions for (23) are obtained from the computed boundary motion.Equation (23) is discretized by finite volume method, and the resulting matrix is solved iteratively using the Algebraic Multigrid (AMG) solver. The cell centered solution for the displacement velocity u
¯ from (23) is interpolated onto the nodes using inverse distance weighted averaging, and the node positions are updated according to [17]:
(25)
x
¯
new
=
x
¯
old
+
u
¯
Δ
t
.For (25), Δ
t is one coupling step that includes 1~3 iterations and one iteration includes 30 fluid solver iteration steps which make sure every coupling iteration step is converged. Use this method, decreasing the diffusivity in larger cells causes those cells to absorb more of mesh motion. It is better to preserve the cell quality of smaller cells which are close to the boundary motion and therefore preserve the computational accuracy.


## 2.3.1. Geometry Model and Initial Conditions

The model used in this work is a wrap fins rocket with a slenderness ratio of more than 25. For comparing with the aerodynamic loads distribution of the elastic slender rocket, it is divided into many segments along the rocket body. Then, through computation, the aerodynamic parameters of every segment are obtained. The slender rocket consists of three parts as shown in Figure2(a). Longitudinally, the body is divided into 31 segments as tabulated in Table 1. The middle point of each section is the normal force action point. The geometry model of the rocket and the flow control volume (see Figure 2(b)) are established. The far-field should be placed far enough from the rocket since the free stream of infinity is defined as boundary conditions. For supersonic computation, the rocket has the far-field boundary located at about 5
D (where D is the rocket diameter) upstream away from the rocket-nose, about 30
D away from the rocket-bottom downstream, and about 15
D away from the rocket-surface in a lateral (see Figure 2(b)).Table 1
Rocket segmentation.
Part 1 (x
1
/
D)
1
2
2.3
Part 2 (x
2
/
D)
0.2
0.4
0.6
0.8
1
1.5
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Part 3 (x
3
/
D)
2Model geometry and flow control volume: (a) longitudinal division of the slender rocket and (b) flow control volume.
(a)
(b)The aerodynamic parameters distribution of the rigid rocket are calculated in the conditions that the angle of attacks is 2° and 4°, while the Mach numbers are 1.5, 2, 2.5, and 3, respectively. The rocket three parts maintain the defined density, Poisson’s ratio, and Young’s modulus, respectively (see Table2).Table 2
Material properties for rocket.
Part name
Density
Young’s modulus
Poisson’s ratio
Part 1, Part 3
7850
2
E
+
11
0.3
Part 2
2770
7.1
E
+
10
0.33


## 2.3.2. CFD Governing Equations

For CFD problems, Direct Numerical Simulation (DNS) solves the Navier-Stokes (N-S) equations directly without the turbulent model needing high speed and large memory computer due to their high nonlinearity and complexity. It is impossible to adopt this method in practical engineering. Decomposing the N-S equations into the RANS equations makes it possible to simulate engineering fluid dynamic problems.Reynolds averaged N-S equations (RANS) can be expressed as(1)
∂
Q
∂
t
+
∂
E
∂
x
+
∂
F
∂
y
+
∂
G
∂
z
=
∂
E
v
∂
x
+
∂
F
v
∂
y
+
∂
G
v
∂
z
,
where
(2)
Q
=
[
ρ
,
ρ
u
,
ρ
v
,
ρ
w
,
ρ
E
]
T
,
E
=
[
ρ
u
,
ρ
u
2
+
p
,
ρ
u
v
,
ρ
u
w
,
(
ρ
E
+
p
)
u
]
T
,
F
=
[
ρ
v
,
ρ
v
u
,
ρ
v
2
+
p
,
ρ
v
w
,
(
ρ
E
+
p
)
v
]
T
,
G
=
[
ρ
w
,
ρ
w
u
,
ρ
w
v
,
ρ
w
2
+
p
,
(
ρ
E
+
p
)
w
]
T
,
E
v
=
[
0
,
τ
x
x
,
τ
x
y
,
τ
x
z
,
β
x
]
T
,
F
v
=
[
0
,
τ
y
x
,
τ
y
y
,
τ
y
z
,
β
y
]
T
,
G
v
=
[
0
,
τ
z
x
,
τ
z
y
,
τ
z
z
,
β
z
]
T
,
where ρ is the fluid density, (
u
,
v
,
w
) are the three components of the Cartesian coordinate system, p is pressure, and E is total energy of unit mass.N-S equations of rotating coordinates [9] are
(3)
∂
ρ
∂
t
+
∂
∂
x
j
(
ρ
w
j
)
=
0
,
∂
∂
t
(
ρ
u
i
)
+
∂
∂
x
j
(
ρ
w
j
w
i
+
p
δ
i
j
)
=
∂
∂
x
j
(
τ
i
j
)
+
ρ
F
i
,
∂
∂
t
(
ρ
E
′
)
+
∂
∂
x
j
(
ρ
w
j
E
′
+
w
j
p
)
=
∂
∂
x
j
(
w
i
τ
i
j
+
k
∂
T
∂
x
j
)
+
ρ
q
-
+
ρ
f
i
w
i
,
where
(4)
τ
i
j
=
μ
[
(
∂
w
i
∂
x
j
+
∂
w
j
∂
x
i
)
-
2
3
δ
i
j
∂
w
k
∂
x
k
]
,
F
i
=
f
i
-
2
e
i
j
k
ω
j
w
k
+
∂
∂
x
j
(
ω
2
R
2
2
)
,
E
′
=
e
+
W
2
2
-
ω
2
R
2
2
=
P
ρ
(
γ
-
1
)
+
W
2
2
-
ω
2
R
2
2
,
where W is the relative speed, E
′ is the total energy of relative rotation, and q
- is the radiant heat (q
-
=
0 in the present study). F is the external force, including volume force and centrifugal force. ω is the angular velocity of rotation and R is the radius of gyration.For the closure of the above equations, the shear stress transport (SST)k
-
ω turbulence model developed by Menter [13, 14] k
-
ε and standard k
-
ω model boundary layer inside and outside. k and ω transport equation can be written as [14]
(5)
d
(
ρ
k
)
d
t
=
τ
i
j
∂
u
i
∂
x
j
-
β
*
ρ
ω
k
+
∂
∂
x
j
[
(
μ
+
σ
k
μ
t
)
∂
k
∂
x
j
]
,
d
(
ρ
ω
)
d
t
=
γ
ρ
μ
t
τ
i
j
∂
u
i
∂
x
j
-
β
ρ
ω
2
+
∂
∂
x
j
[
(
μ
+
σ
ω
μ
t
)
∂
ω
∂
x
j
]
+
2
ρ
(
1
-
F
1
)
σ
ω
2
1
ω
∂
k
∂
x
j
∂
ω
∂
x
j
,
where τ
i
j is the shear stress:
(6)
τ
i
j
=
μ
t
(
∂
u
i
∂
x
j
+
∂
u
j
∂
x
i
-
2
3
∂
u
k
∂
x
k
δ
i
j
)
-
2
3
ρ
k
δ
i
j
.
Blending function F
1is
(7)
F
1
=
tanh
(
ar
g
1
4
)
,
where
(8)
ar
g
1
=
min
⁡
[
max
⁡
(
k
0.09
ω
y
,
500
υ
y
2
ω
)
,
4
ρ
σ
ω
2
k
C
D
k
ω
y
2
]
,
C
D
k
ω
=
max
⁡
(
2
ρ
σ
ω
2
1
ω
∂
k
∂
x
j
∂
ω
∂
x
j
,
1
0
-
20
)
.
The eddy viscosity is defined as
(9)
μ
t
=
ρ
α
1
k
max
⁡
(
α
1
ω
,
Ω
F
2
)
,
where Ω is the absolute value of the vorticity.F
2 is given by
(10)
F
2
=
tanh
(
ar
g
2
2
)
,
where ar
g
2
=
max
⁡
(
2
k
/
0.09
ω
y
,
500
μ
/
ρ
y
2
ω
).The constantsϕ of the SST model are calculated from the constants, ϕ
1
,
ϕ
2 as follows:
(11)
ϕ
=
F
1
ϕ
1
+
(
1
-
F
1
)
ϕ
2
,
where set 1
(
ϕ
1
) is the constants of k
-
ω model and set 2
(
ϕ
2
) is the constants of k
-
ε model.The constants of set1
(
ϕ
1
) are
(12)
σ
k
1
=
0.5
,
σ
ω
1
=
0.5
,
β
1
=
0.075
,
β
*
=
0.09
,
κ
=
0.41
,
γ
1
=
(
β
1
β
*
)
-
(
σ
ω
1
κ
2
β
*
)
.
The constants of set 2
(
ϕ
2
) are
(13)
σ
k
2
=
1.0
,
σ
ω
1
=
0.856
,
β
2
=
0.0828
,
β
*
=
0.09
,
κ
=
0.41
,
γ
2
=
(
β
2
β
*
)
-
(
σ
ω
2
κ
2
β
*
)
.
All other parameters are given in [14].In this work, the far-field free stream condition is standard temperature and pressure (101.325 kPa, 288 K) and the far-field boundary is based on Riemann invariants reflecting boundary conditions. The single reference frame (SRF) model is used to perform the problem of spinning rocket. The air is assumed as an ideal gas and the viscosity varies with temperature in accordance with Sutherland three coefficient formulas. The solution method is implicit formulation and Advection Upstream Splitting Method (AUSM). AUSM which is based on the finite volume method and is used to Spatial discrete the space of fluid field. Second Order Upwind scheme is used on convection and turbulent viscosity terms. Flow field mesh number is 800000,Y
+
=
30
-
100 [15]. Flow field computation grid is shown in Figures 3(a) and 3(b).Discrete mesh: (a) flow field grids and (b) the section of grids.
(a)
(b)Solving the above equations, density, velocity, pressure, and so forth could be computed at every grid node. Then, normal, axial, lift, drag, pitching moment, and center of the pressure coefficients can be obtained by following equations, respectively:(14)
C
N
=
N
q
∞
S
,
(15)
C
A
=
A
q
∞
S
,
(16)
C
L
=
C
N
×
cos
⁡
α
-
C
A
×
sin
α
,
(17)
C
D
=
C
N
×
sin
α
+
C
A
×
cos
⁡
α
,
(18)
C
m
=
pitching
moment
q
∞
S
l
,
(19)
x
c
p
=
x
c
g
-
(
C
m
C
N
)
,
where q
∞, α, S, x
c
p, and x
c
g are the dynamic pressure, angle of attack, reference area, center of pressure location, and center of gravity location, respectively. l is length of the rocket. The vertex of the warhead is the reference point.


## 2.3.3. Static Structure Analysis Equations

A static structural analysis determines the displacements, stresses, strains, and forces in the structures or components caused by loads that do not induce significant inertia and damping effects. Steady loading and response conditions are assumed; that is, the loads and the structure’s response are assumed to vary slowly with respect to time. This is an isotropic linear elastic problem. The static structure analysis equation can be written as(20)
[
K
]
{
δ
}
=
{
F
}
,
where [
K
] is stiffness matrix, {δ} is displacement vector, and {
F
} is force vector.To acquire the high quality mesh of the rocket, 10 nodes tetrahedron element (SOLID187) which is used for the wrap fins and warhead and 20 nodes hexahedron element (SOLID186) which is used for simple geometric part are used to perform the structural analysis. For these linear elastic structural mechanics problems,K is a constant and aerodynamic force F is calculated by CFD code. Then, the deformation δ can be obtained. The rocket solid region grid is shown in Figure 4.Figure 4
Solid region grid.Multipoint Constraint (MPC) approach is used to define the bonding of contact surfaces. Normal force and tangential force of each part are transferred through contact surfaces. Inertia relief method, which is based on d’Alembert principle and used to simulate unconstrained rockets in flight and make sure the rocket has no rigid body displacement, is used with the linear solver. In addition, because the body is rotating, a rotating angular velocity is applied to the rocket body.Inertia relief is an approach in which the applied forces and torques are balanced by inertial forces induced by an acceleration field. Consider the application of an acceleration field (to be determined) that precisely balances the applied loads:(21)
{
F
t
a
}
+
[
M
t
]
{
a
t
1
}
=
{
0
}
,
{
F
r
a
}
+
[
M
r
]
{
a
r
1
}
=
{
0
}
,
where{
F
t
a
} = force components of the applied load vector,
{
F
r
a
} = moment components of the applied load vector,
{
a
t
1
} = translational acceleration vector due to inertia relief (to be determined),
{
a
r
1
} = rotational acceleration vector due to inertia relief (to be determined),
{
r
} = position vector = [
x
,
y
,
z
]
T,
[
M
t
] = mass tensor for the entire finite element model (developed below),
[
M
r
] = mass moments and mass products of the inertia tensor for the entire finite element model (developed below).Once [
M
t
] and [
M
r
] are developed and {
F
t
a
} and {
F
r
a
} can be obtained from CFD output, then {
a
t
1
} and {
a
r
1
} in (21) can be solved. The output inertia relief summary includes {
a
t
1
} (output as translational accelerations) and {
a
r
1
} (output as rotational accelerations).


## 2.3.4. Coupling Boundary Conditions and Dynamic Mesh

In this fluid-structure interaction work, there is a boundary where the fluid and structural domains interactions occur at this boundary or interface. Deformation compatibility conditions and force equilibrium conditions should be satisfied on the fluid-structure coupling interface as [16]
(22)
d
f
=
d
s
,
n
·
τ
f
=
n
·
τ
s
,
q
f
=
q
s
,
T
f
=
T
s
,
where d, q, T, and τ are displacement, heat flux, temperature, and stress field on the fluid-structure coupling interface, respectively. n is the normal direction of interface. Subscripts fand s represent the fluid and the solid, respectively.For data transfer of the dissimilar meshes, the interpolation operation is an essential step. The interpolation called globally conservative is used in this paper. It collects the interface meshes from the CFD code, does the mapping, and communicates stagger loop controls to the CFD code. The mapping generated by ANSYS code is used to interpolate loads between dissimilar meshes on either side of the coupling interface. Each field solver advances through a sequence of multifield stagger (coupling) iterations within each coupling step. During every stagger iteration, each field solver collects the loads that it requires from the other field solvers and then solves its physics fields.The flow field grids motion is achieved by diffusion-based smoothing method which is based on cell volume. And the mesh motion is governed by the diffusion equation:(23)
∇
·
(
γ
∇
u
¯
)
=
0
,
where u
¯ is the mesh displacement velocity and γ is the diffusion coefficient
(24)
γ
=
1
V
α
,
where V is the normalized cell volume and α is the user input parameter (α
=
1.9 in the present study). The boundary conditions for (23) are obtained from the computed boundary motion.Equation (23) is discretized by finite volume method, and the resulting matrix is solved iteratively using the Algebraic Multigrid (AMG) solver. The cell centered solution for the displacement velocity u
¯ from (23) is interpolated onto the nodes using inverse distance weighted averaging, and the node positions are updated according to [17]:
(25)
x
¯
new
=
x
¯
old
+
u
¯
Δ
t
.For (25), Δ
t is one coupling step that includes 1~3 iterations and one iteration includes 30 fluid solver iteration steps which make sure every coupling iteration step is converged. Use this method, decreasing the diffusivity in larger cells causes those cells to absorb more of mesh motion. It is better to preserve the cell quality of smaller cells which are close to the boundary motion and therefore preserve the computational accuracy.


## 3. Presentation of Result

Firstly, the aerodynamic coefficients of the rigid nonspinning rocket are calculated in the conditions of different Mach numbers, and the results are compared with the experimental data, which verified the accuracy of CFD output as shown in Figures5(a), 5(b), and 5(c).Aerodynamics coefficients versus Mach number (Ma): (a) C
D, (b) C
L, and (c) C
m.
(a)
(b)
(c)


### 3.1. Static Aeroelastic Simulation Results of Nonspinning Rocket

Distributed normal force coefficients along rocket length atMa
=
2,3 and α
=
4
∘ of the rocket is shown in Figure 6. It reveals that the normal force of the warhead and tail of the rocket are larger than other parts of the rocket obviously. Then, these applied forces are balanced by inertial forces induced by an acceleration field according to inertial relief method and d’Alembert principle. Therefore, the rocket is bent upward (see Figures 7(a) and 7(b)).Figure 6
Distributed normal force coefficients along rocket length.Y-direction deformation of nonspinning rocket body: (a) Y-direction deformation of nonspinning rocket body at Ma
=
1.5 and (b) Y-direction deformation of nonspinning rocket body at α
=
4
∘.
(a)
(b)One-way coupling is a method which is not considered as the influence of aerodynamic transformation due to elastic structure deformation. Firstly, the structural grid of the flow field around the required model is constructed to compute the aerodynamic pressures distribution along the whole rocket using CFD. Then, map the pressures at the CFD grid points to be replaced by forces on the CSD nodes. Consequently, stress and deformation distributions of the structure are obtained by CSD code. To obtain accurately the deformation and aerodynamic load distributions of the rocket, a two-way coupling method is applied and compared with one-way coupling method. The deformations of the nonspinning rocket, which are calculated by one-way and two-way couplings, are shown in Figure7(a). These deformations are in the same magnitude with one-way coupling output, which is verified by the rationality of the two-way coupling output. The results show that the deformation levels of two-way coupling output are smaller in comparison with one-way coupling. It is due to the normal force of the deformed rocket becoming smaller than the rigid rocket. Obviously, with increasing the Mach number and angle of attack, the deformations of the rocket become larger as presented in Figures 7(a) and 7(b). The total deformation and streamline contour of the rocket at Ma
=
1.5 and α
=
2
∘are shown in Figure 8.Figure 8
Total deformation and streamline contour atMa
=
1.5 and α
=
2
∘.The changes of aerodynamic coefficients with Mach number of elastic rocket compared with the rigid ones are shown in Figures9(a), 9(b), 9(c), and 9(d). It reveals that the drag and lift force coefficients decrease and pitching moment coefficients increase due to elastic deformations. Center of pressure coefficient moves forward (see Figure 9(d)), which contributes to the stability reduction of the rocket.Aerodynamics coefficients versus Mach number (Ma) for elastic and rigid rocket: (a) C
D, (b) C
L, (c) C
m, and (d) X
c
p.
(a)
(b)
(c)
(d)


### 3.2. Static Aeroelastic Simulation Results of Spinning Rocket

Distributed normal force coefficient along rocket length atMa
=
2,3 and α
=
4
∘of spinning elastic rocket is obtained and compared with a rigid one; result is shown in Figure 10. Obviously, the normal force in the warhead and tail of spinning rocket are higher than the cylinder part. For spinning rocket, the deformations in both Y- and Z-directions are illustrated in Figures 11(a) and 11(b). Due to the lateral force generated by spinning, the deformation in Z-direction is slightly high. Therefore, spinning and two-way coupling method should be considered carefully in the design stages of the rocket.Figure 10
Spinning rocket distributed normal force coefficient along rocket length atMa
=
2
,
3 and α
=
4
∘.Deformations of spinning rocket body in: (a)Y-direction and (b) Z-direction.
(a)
(b)Total deformation distribution of the nonspinning and spinning rockets is shown in Figures12(a) and 12(b). The maximum deformation occurred in the wrap fins due to large aerodynamic loads. Elastic and rigid wrap fins pressure distributions of the intersecting line of spinning rocket are shown in Figure 13. The result shows that the deformation induced a significant effect on the aerodynamic characteristics of the wrap fin.Total deformation distribution atMa
=
3andα
=
4
∘of the rocket: (a) nonspinning and (b) spinning.
(a)
(b)Figure 13
Pressure versusx-position on the intersecting line of y
=
0.3 m section of the wrap fin of spinning rocket at Ma
=
3 and α
=
4
∘.Comparison of the aerodynamic parameters for rigid and elastic rockets is given in Table3. Results of drag, lift, pitching moment, and pressure center coefficients results for nonspinning and spinning elastic rocket compared with nonspinning and spinning rigid rocket and the result show that the aerodynamic coefficients are significantly affected by the static aeroelastic phenomenon via two-way coupling. The results show that lift and drag force coefficients are decreased by 1.8~7.6% and 0.06~9.3%, respectively. Pitching moment coefficients are increased by 0.2~14.3% and the center of pressure location of elastic rocket moves forward by 0.4~5.8%. In other words, the stability of the rocket is decreased by 0.4~5.8%. That means elastic deformations contribute to the stability reduction of the rocket.Table 3
Total rocket aerodynamic coefficients.
Mach number
Rocket state
C
D
C
L
C
m
X
cp
2 Ma
Nonspinning rigid rocket
0.5967
0.6902
−0.5112
0.700
Nonspinning elastic rocket
0.5764
0.6517
−0.4799
0.695
Spinning rigid rocket
0.5673
0.3209
−0.1654
0.459
Spinning elastic rocket
0.5567
0.3207
−0.1650
0.457
3 Ma
Nonspinning rigid rocket
0.4759
0.5877
−0.3831
0.618
Nonspinning elastic rocket
0.4396
0.5332
−0.3285
0.582
Spinning rigid rocket
0.4608
0.4009
−0.2101
0.486
Spinning elastic rocket
0.4467
0.4001
−0.2096
0.484


## 3.1. Static Aeroelastic Simulation Results of Nonspinning Rocket

Distributed normal force coefficients along rocket length atMa
=
2,3 and α
=
4
∘ of the rocket is shown in Figure 6. It reveals that the normal force of the warhead and tail of the rocket are larger than other parts of the rocket obviously. Then, these applied forces are balanced by inertial forces induced by an acceleration field according to inertial relief method and d’Alembert principle. Therefore, the rocket is bent upward (see Figures 7(a) and 7(b)).Figure 6
Distributed normal force coefficients along rocket length.Y-direction deformation of nonspinning rocket body: (a) Y-direction deformation of nonspinning rocket body at Ma
=
1.5 and (b) Y-direction deformation of nonspinning rocket body at α
=
4
∘.
(a)
(b)One-way coupling is a method which is not considered as the influence of aerodynamic transformation due to elastic structure deformation. Firstly, the structural grid of the flow field around the required model is constructed to compute the aerodynamic pressures distribution along the whole rocket using CFD. Then, map the pressures at the CFD grid points to be replaced by forces on the CSD nodes. Consequently, stress and deformation distributions of the structure are obtained by CSD code. To obtain accurately the deformation and aerodynamic load distributions of the rocket, a two-way coupling method is applied and compared with one-way coupling method. The deformations of the nonspinning rocket, which are calculated by one-way and two-way couplings, are shown in Figure7(a). These deformations are in the same magnitude with one-way coupling output, which is verified by the rationality of the two-way coupling output. The results show that the deformation levels of two-way coupling output are smaller in comparison with one-way coupling. It is due to the normal force of the deformed rocket becoming smaller than the rigid rocket. Obviously, with increasing the Mach number and angle of attack, the deformations of the rocket become larger as presented in Figures 7(a) and 7(b). The total deformation and streamline contour of the rocket at Ma
=
1.5 and α
=
2
∘are shown in Figure 8.Figure 8
Total deformation and streamline contour atMa
=
1.5 and α
=
2
∘.The changes of aerodynamic coefficients with Mach number of elastic rocket compared with the rigid ones are shown in Figures9(a), 9(b), 9(c), and 9(d). It reveals that the drag and lift force coefficients decrease and pitching moment coefficients increase due to elastic deformations. Center of pressure coefficient moves forward (see Figure 9(d)), which contributes to the stability reduction of the rocket.Aerodynamics coefficients versus Mach number (Ma) for elastic and rigid rocket: (a) C
D, (b) C
L, (c) C
m, and (d) X
c
p.
(a)
(b)
(c)
(d)


## 3.2. Static Aeroelastic Simulation Results of Spinning Rocket

Distributed normal force coefficient along rocket length atMa
=
2,3 and α
=
4
∘of spinning elastic rocket is obtained and compared with a rigid one; result is shown in Figure 10. Obviously, the normal force in the warhead and tail of spinning rocket are higher than the cylinder part. For spinning rocket, the deformations in both Y- and Z-directions are illustrated in Figures 11(a) and 11(b). Due to the lateral force generated by spinning, the deformation in Z-direction is slightly high. Therefore, spinning and two-way coupling method should be considered carefully in the design stages of the rocket.Figure 10
Spinning rocket distributed normal force coefficient along rocket length atMa
=
2
,
3 and α
=
4
∘.Deformations of spinning rocket body in: (a)Y-direction and (b) Z-direction.
(a)
(b)Total deformation distribution of the nonspinning and spinning rockets is shown in Figures12(a) and 12(b). The maximum deformation occurred in the wrap fins due to large aerodynamic loads. Elastic and rigid wrap fins pressure distributions of the intersecting line of spinning rocket are shown in Figure 13. The result shows that the deformation induced a significant effect on the aerodynamic characteristics of the wrap fin.Total deformation distribution atMa
=
3andα
=
4
∘of the rocket: (a) nonspinning and (b) spinning.
(a)
(b)Figure 13
Pressure versusx-position on the intersecting line of y
=
0.3 m section of the wrap fin of spinning rocket at Ma
=
3 and α
=
4
∘.Comparison of the aerodynamic parameters for rigid and elastic rockets is given in Table3. Results of drag, lift, pitching moment, and pressure center coefficients results for nonspinning and spinning elastic rocket compared with nonspinning and spinning rigid rocket and the result show that the aerodynamic coefficients are significantly affected by the static aeroelastic phenomenon via two-way coupling. The results show that lift and drag force coefficients are decreased by 1.8~7.6% and 0.06~9.3%, respectively. Pitching moment coefficients are increased by 0.2~14.3% and the center of pressure location of elastic rocket moves forward by 0.4~5.8%. In other words, the stability of the rocket is decreased by 0.4~5.8%. That means elastic deformations contribute to the stability reduction of the rocket.Table 3
Total rocket aerodynamic coefficients.
Mach number
Rocket state
C
D
C
L
C
m
X
cp
2 Ma
Nonspinning rigid rocket
0.5967
0.6902
−0.5112
0.700
Nonspinning elastic rocket
0.5764
0.6517
−0.4799
0.695
Spinning rigid rocket
0.5673
0.3209
−0.1654
0.459
Spinning elastic rocket
0.5567
0.3207
−0.1650
0.457
3 Ma
Nonspinning rigid rocket
0.4759
0.5877
−0.3831
0.618
Nonspinning elastic rocket
0.4396
0.5332
−0.3285
0.582
Spinning rigid rocket
0.4608
0.4009
−0.2101
0.486
Spinning elastic rocket
0.4467
0.4001
−0.2096
0.484


## 4. Conclusions

In this paper, in order to solve the static aeroelastic problem, a two-way coupling FSI method is introduced. ANSYS Workbench multiphysics coupling platform is used to determine the static aeroelastic behavior of the slender rocket. Diffusion-based smoothing method and inertia relief methods are used to calculate the static deformations and aerodynamic characteristics of the deformed rocket. With increasing the Mach number and angle of attack, the deformations of the rocket become large obviously. For spinning rocket,Z-direction is also deformed due to the lateral force generated by the spinning. The principle of the bending direction of the rocket is explained in this paper. The aerodynamic characteristics and stability of the rocket are changed significantly due to deformations. Consequently, spinning and two-way coupling should be considered carefully during the design.



---

*Source: 478534-2014-02-17.xml*