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Supersonic flow study of a capsule geometry using large-eddy simulation
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Supersonic flow study of a capsule geometry using large-eddy simulation
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Content
Supersonic Flow Study of a Capsule Geometry using Large-Eddy
Simulation
by
Junhao Ma
A Thesis Presented to the
FACULTY OF THE USC VITERBI SCHOOL OF ENGINEERING
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF SCIENCE
AEROSPACE ENGINEERING
May 2023
Copyright 2023 Junhao Ma
ii
Acknowledgments
I would like to acknowledge my academic advisor, Prof. Carlos Pantano-Rubino, for his guid-
ance throughout my graduate studies. I would also like to thank Prof. Ivan Bermejo-Moreno and
Prof. Mitul Luhar for being on my defense committee. I am also grateful to Hang Yu for his valu-
able assistance on my thesis work. To all persons I acknowledge, thank you for your support in
making this academic year a rewarding experience.
iii
Table of Contents
Acknowledgments........................................................................................................................... ii
List of Tables .................................................................................................................................. v
List of Figures ................................................................................................................................ vi
Nomenclature ............................................................................................................................... viii
Abstract ........................................................................................................................................... x
Chapter 1: Introduction ................................................................................................................... 1
1. 1. Fluid System ..................................................................................................................... 2
1. 2. Computational Fluid Dynamics ........................................................................................ 3
Chapter 2: Capsule Geometry Design............................................................................................. 5
Chapter 3: Computational Models and Approach .......................................................................... 7
3. 1. Large-Eddy Simulation ..................................................................................................... 9
3. 2. Subgrid-Scale Model ...................................................................................................... 13
3. 3. Immersed Boundary Method and Numerical Method Brief ........................................... 16
3. 4. Fluid Domain and Boundary Conditions ........................................................................ 18
Chapter 4: Analysis of Aerodynamic Parameters ......................................................................... 22
4. 1. Computation Error .......................................................................................................... 22
4. 2. Numerical Analysis on Aerodynamic Parameters .......................................................... 24
Chapter 5: Flow Visualization and Analysis ................................................................................ 34
Chapter 6: Conclusion................................................................................................................... 48
References ..................................................................................................................................... 49
Appendices .................................................................................................................................... 51
Appendix 1: Results and Errors (α = 0° ) ................................................................................ 51
iv
Appendix 2: Global Maximum and Minimum Flow Properties at M = 2.5 ........................... 53
Appendix 3: Unsteady L/D Results at M = 2.5 ....................................................................... 54
Appendix 4 (a): Slice View of the Mach Number at M = 2.5 ................................................. 56
Appendix 4 (b): Slice View of the Total Energy (kJ) at at M = 2.5........................................ 58
Appendix 5: U.S. Standard Atmosphere Air Properties - SI Units ......................................... 60
v
List of Tables
Table 1. Mach number range .................................................................................................... 2
Table 2. Grid dimensions of refinement levels ....................................................................... 19
Table 3. US standard atmosphere property at h = 20,000 m................................................... 21
Table 4. Mesh Setup of the Fluid and Solid............................................................................ 23
Table 5. Maximum solution error at 0.1 < t ≤ 0.2 for α = 0° .................................................. 24
Table 6. Average force results at 0.2 < t ≤ 0.25 s ................................................................... 31
Table 7. Aerodynamic parameter results at at 0.2 < t ≤ 0.25 s ............................................... 32
Table 8. Leading shock distance to the capsule base .............................................................. 36
Table 9. Flow properties at the resolved layers ...................................................................... 38
vi
List of Figures
Figure 1. 1 Schematic sketch of a space mission ..................................................................... 1
Figure 1. 2 Capture of the shock wave and wake of a supersonic bullet ................................. 2
Figure 1. 3 Hot layer surrounding the capsule ......................................................................... 3
Figure 2. 1 Nomenclature of the capsule ................................................................................. 5
Figure 2. 2 Capsule design drawing (plane view) .................................................................... 6
Figure 3. 1 Angle of attack (AOA) and the capsule ................................................................ 8
Figure 3. 2 Plane view of the capsule shoulder immersing into the fluid cell ....................... 17
Figure 3. 3 Computational domain and a slice view of mesh to the x-y plane ...................... 19
Figure 3. 4 (a) AMR cells pack; (b) AMR patches at the wake ................................................. 20
Figure 3. 5 Slice view of AMR patches at a shock wave ...................................................... 20
Figure 4. 1 Closeup of a slice view of the x-z plane Mach number contour and AMR
refinement through the center of the computational fluid domain ......................................... 22
Figure 4. 2 Drag results by three levels of mesh quality at α = 0° ......................................... 23
Figure 4. 3 Drag results at M = 2.5 (a) & (b) ......................................................................... 25
Figure 4. 4 Lift (Fy) results at M = 2.5 (a) & (b) ................................................................... 27
Figure 4. 5 Fz results at M = 2.5 (a) & (b).............................................................................. 29
Figure 4. 6 Aerodynamic parameters ..................................................................................... 33
Figure 5. 1 Schematic diagram of the flow compressibility at center x-y plane at α = 0° ..... 34
Figure 5. 2 Three-dimensional perspective of ∇Ux contour .................................................. 35
Figure 5. 3 Slice view of center x-z plane by the Mach number ........................................... 35
Figure 5. 4 Slice view of center x-y plane by the pressure (kPa) .......................................... 37
vii
Figure 5. 5 Slice view of center x-y plane by the Mach number with Mmin = 2.0 ................. 38
Figure 5. 6 Shear layers captured around the capsule by ∇Ux with ∇Ux max = 1500 s
-1
......... 38
Figure 5. 7 Slice view at the capsule through center x-y plane of the compressibility ......... 40
Figure 5. 8 Wake features decomposition of center x-z plane Mach field ............................ 40
Figure 5. 9 Slice view of flow variables at the turbulent wake through center x-y plane ..... 43
Figure 5. 10 Closeup slice view at center x-y plane of AOAs by the Mach number for
α = 0°, α = 2° , α = 5° and α = 10° at t = 0.248 s ..................................................................... 45
Figure 5. 11 Closeup view of central x-y plane at the capsule shoulder by unit velocity
vector....................................................................................................................................... 46
viii
Nomenclature
Acronyms
AMR = Adaptive Mesh Refinement
AOA = Angle of Attack
CFD = Computational Fluid Dynamics
IBM = Immersed Boundary Method
LES = Large-Eddy Simulation
SGS = Subgrid-Scale Model
Symbols
α = AOA
β = Ballistic coefficient
γ = Heat capacity ratio
δij = Kronecker delta
κ = Thermal conductivity
µ = Dynamic viscosity
ν = Kinematic viscosity
ξ = Lagrangian coordinate
фф = Arbitrary field for convolution
ρ = Air density
σ = Total stress tensor
τ = Viscous stress tensor
CL = Lift coefficient
CD = Drag coefficient
Cp = Specific heat constant at constant pressure
D = Drag force
E = Total energy per unit mass
fi = Force function
ix
𝑓 ̃
= Favre filtering operator
G = Convolution kernel
L/D = Lift-drag ratio
h = Geopotential altitude above sea level
H = Heat energy
M = Mach number
P = Absolute pressure
Pr = Prandtl number
q = Heat flux
Re = Reynolds number
R = Molar gas constant
S = Strain-rate tensor
T = Temperature
U = Velocity
x = Eulerian coordinate
Δ = Grid width
∇ = Gradient operator
x
Abstract
This study presents a flow simulation of a 3.89 m diameter capsule at Mach 2.5 dedicated to
investigating various aspects of its flow metrics. The high Reynolds number flow exceeding 10
7
results in strong separation of the boundary layer producing a complicated wake with high dissi-
pation energy. With proper assumptions made, the study utilizes Large-Eddy Simulation (LES)
using a compressible subgrid-scale (SGS) model coupled with the Immersed Boundary Method
(IBM) for cheaper computation approach simulating flight attitudes at 0° , 2° , 5° and 10° angle of
attack. Air properties are presumed at 20,000 m altitude for the freestream flow along with proper
boundary conditions and initializations for the fluid field. The solution fields obtained provide an
insight into the compressible flow behavior over the capsule that leading shock waves are success-
fully captured for each model. Recompression shocks and vortex shedding effects are also caught
along the strong shear layers produced by the capsule geometry. Averaged numerical results indi-
cate minor compressibility change within the AOAs being studied, though significant differences
of wake behaviors are determined. Increasing values from 0° − 10° AOAs changes pressure force
distribution with over 400 kN drag forces and lift-drag ratios ranging from 0 − 0.127.
1
Chapter 1: Introduction
The capsule is a critical part of a spacecraft life system for astronauts that experiences various
stages during a space mission, involving ascent through the atmosphere, space operations, and
planetary entry. A space transportation trajectory includes acceleration, maintenance of velocity,
and deceleration. Advances in scientific study and engineering have enabled spacecraft becoming
more sophisticated and capable of handling challenging missions, requiring meticulous planning
and design to ensure astronauts’ safety and success: a capsule must withstand extreme physical
conditions.
Figure 1. 1 Schematic sketch of a space mission [18]
After completing missions in space, the capsule separates and begins a re-entry, decelerating and
entering the Earth’s atmosphere at a certain trajectory carrying out high energy exchanges that the
capsule’s deceleration motion converts extreme kinetic energy and potential energy to heat. While
a typical atmospheric re-entry involves multiple speed phases, this study is particularly interested
in the capsule’s interaction with supersonic flow. The major objective is to get an insight of a
capsule’s aerodynamic behavior associated with supersonic flow by simulating corresponding re-
entry state through a computational approach.
2
1. 1. Fluid System
Table 1. Mach number range
< 0.8 Subsonic
0.8 − 1.2 Transonic
1 − 5 Supersonic
5 − 10 Hypersonic
> 10 High Hypersonic
The Mach number is a criterion of flow compressibility defined as 𝑀 = 𝑈 /𝑎 , where 𝑈 denotes
the flow velocity and 𝑎 = √𝛾𝑅𝑇 denotes the speed of sound. The speed of sound is a function of
the heat capacity ratio γ, the molar gas constant R, and the temperature T.
Figure 1. 2 Capture of the shock wave and wake of a supersonic bullet [15]
For M∞ > 1 at freestream, shockwaves are expected around the capsule. When air particles
travel through a shock wave, they experience strong compression with sudden increase in density,
pressure and temperature, and decrease in velocity. Because of flow path disturbance, the capsule
also generates boundary layers and complex patterns of flow structures downstream called super-
sonic wake. In addition, as shown in Figure 1. 2, recompression shock may occur due to the need
to straighten the flow after deflection by the main bow shock.
3
Figure 1. 3 Hot layer surrounding the capsule
1. 2. Computational Fluid Dynamics
In CFD, the fluid domain is constructed by a finite number of cells into a mesh set, where
discretized partial differential equations (PDE) approximating nonlinear equations for fluid mo-
tions are solved at every cell. CFD enables researchers to find numerical solutions through appro-
priate governing equations and a fine enough mesh. In particular, the unsteady Navier–Stokes
equations are commonly used in simulations for solving the time-dependent behavior of flows.
The equations describe viscous Newtonian fluid motion of a continuum by accompanying conti-
nuity
1
, conservation of momentum, and conservation of energy relating to flow density, velocity,
pressure and temperature.
One can predict the flow’s complexity by calculating the capsule’s Reynolds number, 𝑅𝑒 =
𝜌 ∞
𝑈 ∞
𝐿 /𝜇 , which measures a fluid’s inertia with respect to its viscosity. 𝜌 ∞
denotes the
freestream density, 𝑈 ∞
denotes the freestream velocity, 𝐿 denotes the characteristic linear
1
Conservation of Mass.
4
dimension
2
and the dynamic viscosity μ [N· s/m
2
] is tangential force per unit area reflecting fluid
deformation. For the capsule in this thesis, the Reynolds number reaching 10
7
indicates the un-
steady flow with presence of strong eddies to be computationally time expensive due to its highly
nonlinear nature which contains numerous small-scale turbulent structures. For a high Reynolds
flow, a large number of eddies in small spatial and temporal scales require sufficient cells matching
in scales
3
for Direct Numerical Simulation (DNS) to attain accurate solution fields that the demand
on computing resources by the DNS is overwhelming.
Large-Eddy Simulation (LES) is used as an alternative computational approach for dealing
with turbulent structures in a cheaper way: the LES solves the Navier-Stokes equations through
incorporating subgrid-scale (SGS) models which replace small-scale eddies. This is valid because
the small scales of a flow are presumably more isotropic and universal to be possibly modeled [2].
Therefore, a well-implemented LES is capable of solving high Reynolds number flow that the SGS
fluid motion can be modeled by a defined resolved field in a fluid solver to enable coarser compu-
tational resolution with occurrence of chaotic turbulence smaller than the cell size.
2
Not equivalent to the Lift (L).
3
i.e. the resolution of a computational field.
5
Chapter 2: Capsule Geometry Design
A capsule's aerodynamic performance depends on its geometry. While this thesis does not
study hypersonic flow, it is important to consider its impact on capsule aerodynamics as it repre-
sents the upper limit of the aerodynamic environment. The geometry of a capsule is characterized
by its length, diameter and body shape. While accompanying the overall length and diameter with
mission requirements such as crew size, payload and fuel capacity, body shape is specifically de-
signed to generate and sustain drag load for deceleration at re-entry state considering the propul-
sion limit of onboard engines.
Figure 2. 1 Nomenclature of the capsule
On the other hand, a launch escape system must be equipped for aborting a launch mission when
necessary. Hence, the capsule is best attached at the top of a transport rocket for safety and effi-
ciency, implying a compatible geometry design shall be applied to adapt high-speed flow encoun-
tered during a launch. The constraints above are summarized into the guidance for the capsule
geometry design:
1). A frustum shaped body structure.
2). A blunt surface facing reentry direction best adapts the re-entry objective.
3). Smooth shape at edges.
6
Figure 2. 2 Capsule design drawing (plane view)
The 3.89 × 4.2 m prototype capsule has a 4-spherical-segment shape, consisting of four circular
geometries for cap, body, and base, in which the base faces the reentry direction. The capsule's
primary feature is a 0.8 m radius spherical cap integrated into the main body structure. This cap is
smoothly connected to the body via a second spherical curve, ensuring a seamless transition. The
body of the capsule features a 19.27° frustum shape with a slight curvature. The capsule base is a
fourth spherical shape, featuring a 5.9 m radius, which allows the capsule to effectively detach
shock wave for avoiding direct heat contact while maximizing drag force during reentry. To further
optimize the flow over the base of the capsule, a 0.25 m radius corner cut at the convex shoulder
is implemented to create smoother flow transfer. The sample geometry is an idealized model for
flow study with ignored cavities for engine nozzle channels being a potential drawback.
7
Chapter 3: Computational Models and Approach
The flow model uses a fluid solver introduced by “Large-Eddy Simulation of Flow Over De-
formable Parachutes using Immersed Boundary and Adaptive Mesh” [1] originally developed for
fluid-structure interaction. The air is treated as a non-reactive mono-species gas and Newtonian
fluid so that there is no existence of combustion, and the dynamic viscosity of the air is a function
of temperature only. It is also assumed that the air is a type of single-phase fluid for supersonic
flow. Dissociation and ionization effects at molecular level and radiative heat transfer are ne-
glected, and local thermodynamic equilibrium in the flow is assumed to be valid [4]. This enables
calorically perfect gas assumption involving usage of constant specific heat Cp = 1004.46 J/kg· K,
heat capacity ratio γair = 1.4 and molar gas constant Rair = 287.058 m
2
/s
2
· K to be independent of
temperature. The Reynolds number Re = 1.80 × 10
7
indicates the presence of large disturbances
caused by the capsule geometry. To compute the supersonic turbulent flow using the LES, the fluid
solver uses spatial-filtered governing equations that incorporate a compressible SGS model to ac-
count for SGS turbulence dynamics as well as compressibility effects, which targets on modeling
the direct kinetic energy cascade from large to small-scales in isotropic turbulence and high-Reyn-
olds fully developed turbulent flows. The truncation capability alleviates calculation cost for tur-
bulence flow due to the large time and length scales of the resolved field [4].
To study the capsule’s aerodynamic behavior, angle of attack (AOA, written as α) is of interest
combining lift (L) and drag force (D) produced to calculate lift coefficient (CL), drag coefficient
(CD) and lift-drag ratio (L/D). α is the angle formed between the free stream flow U∞ and the
capsule’s longitudinal axis formed along the leading edge and the trailing edge. The axisymmetric
capsule geometry produces lift force at non-zero AOAs.
8
Figure 3. 1 Angle of attack (AOA) and the capsule
The drag and lift equations are:
𝐿 =
1
2
𝜌 𝑈 ∞
2
𝐴 𝑟𝑒𝑓
𝐶 𝐿
(1)
𝐷 =
1
2
𝜌 𝑈 ∞
2
𝐴 𝑟𝑒𝑓
𝐶 𝐷
(2)
Where 𝜌 is the air density, U∞ is the freestream velocity, Aref is the reference area of the capsule.
CL and CD are functions of the Mach number and AOA (α). For the capsule, its cross-sectional area
(11.20 m
2
) is chosen for Aref. Model (1) and Model (2) are not used explicitly for force calculations
but establish L/D relation:
𝐿 𝐷 =
𝐶 𝐿 ( 𝑀 , 𝛼 )
𝐶 𝐷 ( 𝑀 , 𝛼 )
(3)
The aerodynamic force exerting on the capsule is mainly pressure force caused by fluid flow form-
ing a high-pressure distribution, mostly contributing to drag ahead of the capsule. The other con-
tribution is the viscous force. Force computation is accomplished by integrating the boundary force
components exerting on the capsule obtained through the immersed boundary method. L and D
obtained are used for calculating CL and CD by Model (1) and Model (2).
9
3. 1. Large-Eddy Simulation
The compressible Navier-Stokes equations are used for solving the Mach 2.5 fluid model. The
total stress tensor 𝜎 𝑖𝑗
[Pa] is an essential concept that describes the local stress for viscid flow,
composing the viscous stress tensor 𝜏 𝑖𝑗
and the hydrostatic pressure. The physical relation of terms
can be written as:
𝜎 𝑖𝑗
= 𝜏 𝑖𝑗
− 𝑝 𝛿 𝑖𝑗
(4)
The pressure term with the kronecker delta δij (δij = 1 if i = j and δij = 0 if i ≠ j) denotes normal
stress components due to pressure. By writing the differential form of the unsteady compressible
Navier-Stokes equation expressing rate change of mass, rate of momentum, and rate of energy, the
governing equations for a fully resolved scale can be formulated by plugging in Model (4), where
the pressure term in 𝜎 𝑖𝑗
reduces to a force term in the momentum equation and the energy equation
in Model (5). Note that to implement the immersed boundary method (IBM), a force function
𝑓 𝑖 ( 𝑥 ; 𝑡 ) for building the boundary conditions of the fluid-structure interaction needs to be derived,
see the brief in Chapter 3. 3.
𝜕𝜌
𝜕𝑡
+
𝜕𝜌 𝑢 𝑗 𝜕 𝑥 𝑗 = 0
𝜕𝜌 𝑢 𝑖 𝜕𝑡
+
𝜕𝜌 𝑢 𝑖 𝑢 𝑗 𝜕 𝑥 𝑗 = −
𝜕𝑝
𝜕 𝑥 𝑖 +
𝜕 𝜏 𝑖𝑗
𝜕 𝑥 𝑗 + 𝑓 𝑖
(5)
𝜕𝜌𝐸 𝜕𝑡
+
𝜕 ( 𝜌𝐸 + 𝑝 ) 𝑢 𝑗 𝜕 𝑥 𝑗 =
𝜕 𝜏 𝑖𝑗
𝑢 𝑖 𝜕 𝑥 𝑗 −
𝜕 𝑞 𝑗 𝜕 𝑥 𝑗 + 𝑢 𝑗 𝑓 𝑗
10
Subscript i and j represent indices of variable components of a coordinate system for simplicity.
The governing equations establish the PDEs involving the density ρ [kg/m
3
], the velocity u [m/s],
the pressure P [Pa], the total energy E [J], the heat flux q [W/m
2
] and the force 𝑓 𝑖 ( 𝑥 ; 𝑡 ) [N], where
the total energy per unit mass E is the summation of the internal and kinetic energy:
𝜌𝐸 =
𝑝 𝛾 − 1
+
1
2
𝜌 𝑢 𝑖 𝑢 𝑖
(6)
The viscous stress tensor 𝜏 𝑖𝑗
, composing shear stress and normal stress, expresses the local defor-
mation motion of the fluid by the strain-rate tensor S [s
-1
]: the first term is a deviatoric part repre-
senting the deformation caused by the shear stress; the second term is a trace part of the strain-rate
tensor representing the normal stress causing compression or tension motion due to the shear stress.
𝜏 𝑖𝑗
is written as:
𝜏 𝑖 𝑗 = 𝜇 ( 𝑇 )(
𝜕 𝑢 𝑖 𝜕 𝑥 𝑗 +
𝜕 𝑢 𝑗 𝜕 𝑥 𝑖 ) −
2
3
𝜇 ( 𝑇 ) (
𝜕 𝑢 𝑘 𝜕 𝑥 𝑘 ) 𝛿 𝑖𝑗
(7)
𝑆 =
1
2
(
𝜕 𝑢 𝑖 𝜕 𝑥 𝑗 +
𝜕 𝑢 𝑗 𝜕 𝑥 𝑖 )
As the pre-assumptions proposed for supersonic flow, the dynamic viscosity is a variable with a
function to the flow temperature by the Sutherland’s law valid from 100 K to 1900 K, where S1 =
110.4 K [8]:
𝜇 ( 𝑇 )
𝜇 ( 𝑇 0
)
= (
𝑇 𝑇 0
)
3
2
𝑇 0
+ 𝑆 1
𝑇 + 𝑆 1
(8)
11
qj denotes heat flux written as:
𝑞 𝑗 = −𝜅 ( 𝑇 )
𝜕𝑇
𝜕 𝑥 𝑗
(9)
𝜅 = 𝜇 ( 𝑇 ) 𝐶 𝑝 𝑃𝑟
(10)
where κ denotes the thermal conductivity to be a function of the dynamic viscosity μ, the specific
heat at constant pressure Cp and the Prandtl number Pr. The Prandtl number
4
measures the ratio
between the kinematic viscosity v = μ/ρ [m
2
/s] and the thermal diffusivity κ/(ρCp) [m
2
/s] [4].
Because the resolved field by Model (5) does not account the effect caused by the small-scale
motions, a method relating to the SGS model is necessary for the formulations. A filtering ap-
proach is used for sorting out the resolved field by assuming an arbitrary field ф(x, t) as a convo-
lution product by kernel G.
𝜙 ̅( x, 𝑡 ) = ∫ 𝜙 ( x, 𝑡 ) 𝐺 ( x − x
′
) 𝑑 𝑥 ′
∞
−∞
(11)
Despite this mathematical expression does not explicitly solve the flow model, it conceptualizes
the relation between the LES resolved field 𝜙 ̅
and the fully resolved field 𝜙 5
. To distinguish the
resolved scales, the LES uses a density-based filtering operator, namely Favre-filtering, which
separates the resolved scales and the subgrid scales for compressible flows:
𝑓 ̃
=
𝜌 𝑓 ̅ ̅ ̅ ̅
𝜌 ̅
(12)
4
Assumed to be constant.
5
i.e., a well-imposed DNS by Model (5) [1].
12
𝑓 ̃
denotes the low frequency part of a variable f. The overbar for 𝜌 ̅ denotes the density obtained
for the resolved scale; The tilde denotes the filtering operation on a variable capturing primary
flow features larger than a prescribed cutoff scale Δ (alternatively called filter width). By applying
the Favre-filtering operator to the Navier-Stokes equations, the fluid motions are supplemented by
the SGS models in the governing equations, where the SGS models are functions of the resolved
scales. In other words, this framework numerically computes the resolved variables while simul-
taneously modeling the subgrid-scale (SGS) motions.
𝜕 𝜌 ̅
𝜕𝑡
+
𝜕 𝜌 ̅ 𝑢 ̃
𝑗 𝜕 𝑥 𝑗 = 0
𝜕 𝜌 ̅ 𝑢 ̃
𝑖 𝜕𝑡
+
𝜕 𝜌 ̅ 𝑢 ̃
𝑖 𝑢 ̃
𝑗 + 𝑝 ̅ 𝛿 𝑖𝑗
𝜕 𝑥 𝑗 −
𝜕 𝜏 ̂
𝑖𝑗
𝜕 𝑥 𝑗 = −
𝜕 𝜏 𝑖𝑗
𝑠𝑔𝑠 𝜕 𝑥 𝑗 + 𝑓 𝑖
(13)
𝜕 𝜌 ̅ 𝐸 ̂
𝜕𝑡
+
𝜕 ( 𝜌 ̅ 𝐸 ̂
+ 𝑝 ̅ ) 𝑢 ̃
𝑗 𝜕 𝑥 𝑗 −
𝜕 𝜏 ̂
𝑖𝑗
𝑢 ̃
𝑖 𝜕 𝑥 𝑗 +
𝜕 𝑞̂
𝑗 𝜕 𝑥 𝑗 = −
𝜕 𝑞 𝑗 𝑠𝑔𝑠 𝜕 𝑥 𝑗 −
𝜕 𝜏 ̂
𝑖𝑗
𝑠𝑔𝑠 𝑢 ̃
𝑖 𝜕 𝑥 𝑗 + 𝑢 ̃
𝑗 𝑓 𝑗
The decomposition is not applied to the pressure field [4]. Its filtered equation of state can be
related by the ideal gas equation where Rair is equal to 287.058 m
2
/s
2
· K:
𝑝 ̅ = 𝜌 ̅ 𝑅 𝑇 ̃
(14)
Note that:
𝜌 𝑢 𝑖 ̅ ̅ ̅ ̅ ̅ = 𝜌 ̅ 𝑢 ̃
𝑖 , 𝜌 𝑢 𝑖 𝑢 𝑗 ̅ ̅ ̅ ̅ ̅ ̅ ̅ = 𝜌 ̅ 𝑢 𝑖 𝑢 𝑗 ̃ , and 𝜌𝑇
̅ ̅ ̅ ̅
= 𝜌 ̅ 𝑇 ̃
13
The Favre-filtering operator and the force term in the governing equations are distinct parameters.
The caret (^) denotes filtered flow variables obtained from the computational field. Vreman estab-
lishes a relation with computable 𝐸 ̂
to left-hand side of the total energy per unit mass with the
filtering operator applied [4][7], though the variables are not modified:
𝜌 ̅ 𝐸 ̂
=
𝑝 ̅
𝛾 − 1
+
1
2
𝜌 ̅ 𝑢 𝑖 𝑢 𝑖 ̃
(15)
Similarly, computable viscous stress tensor and heat flux are:
𝜏 ̂
𝑖𝑗
= 𝜇 ( 𝑇 ̃
)(
𝜕 𝑢 ̃
𝑖 𝜕 𝑥 𝑗 +
𝜕 𝑢 ̃
𝑗 𝜕 𝑥 𝑖 ) −
2
3
𝜇 ( 𝑇 ̃
) (
𝜕 𝑢 ̃
𝑘 𝜕 𝑥 𝑘 ) 𝛿 𝑖𝑗
(16)
𝑞̂
𝑗 = −𝜅 ( 𝑇 ̃
)
𝜕 𝑇 ̃
𝜕 𝑥 𝑗
(17)
The dynamic viscosity μ(𝑇 ̃
) and thermal conductivity κ(𝑇 ̃
) can be calculated by Model (8) and
Model (10) with Favre-filtered temperature.
3. 2. Subgrid-Scale Model
In Model (13), 𝜏 𝑖𝑗
𝑠𝑔𝑠 denotes the SGS stress tensor and 𝑞 𝑖 𝑠𝑔𝑠 denotes heat flux. Their general
forms are defined as the difference between the resolved scale and the total scale:
𝜏 𝑖𝑗
𝑠𝑔𝑠 = ( 𝜌 ̅ 𝑢 𝑖 𝑢 𝑗 ̃ − 𝜌 ̅ 𝑢 ̃
𝑖 𝑢 ̃
𝑗 )− ( 𝜏 ̅
𝑖𝑗
− 𝜏 ̂
𝑖𝑗
)
(18)
𝑞 𝑖 𝑠𝑔𝑠 = ( 𝜌 𝑢 𝑗 𝐸 ̅ ̅ ̅ ̅ ̅ ̅
− 𝜌 ̅ 𝑢 ̃
𝑗 𝐸 ̃
)+ ( 𝑢 𝑗 𝑝 ̅ ̅ ̅ ̅− 𝑢 ̃
𝑗 𝑝 ̅ )− ( 𝜏 𝑖𝑗
𝑢 𝑗 ̅ ̅ ̅ ̅ ̅ ̅ )− ( 𝑞̅
𝑗 − 𝑞̂
𝑗 )
(19)
14
The SGS model is developed based on several basic functional modeling hypotheses [4]
6
, in-
cluding the Boussinesq hypothesis, explicitly or implicitly for viscosity and diffusivity so that:
(1). Little information is neglected.
(2). The energy cascade mechanism is built by a model similar to molecular diffusion.
(3). A total separation exists between the SGS and resolved scales.
(4). Calibration on turbulent kinetic energy production ensures conservation.
Compared to a forward energy transfer toward the SGS scales, a backward energy cascade is much
weaker and negligible [4]. When transferring from the resolved scales to the subgrid scales, the
kinetic energy of the resolved scales converts into SGS kinetic energy and SGS heat energy. In
general, the dissipation of energy caused by viscosity is more significant at smaller scales, where
the rate changes of velocity are higher and the viscous forces are stronger. This term is incorporated
into the SGS modeling, and the characteristics of high velocity gradients are utilized for flux eval-
uation for small scales.
The Smagorinsky model is a widely used SGS modeling for approximating small-scale turbu-
lence developed from Boussinesq hypothesis and eddy viscosity to relate the SGS turbulence to
resolved scale
7
. The Boussinesq hypothesis relates Reynolds stress
8
to the mean flow’s strain rate;
the eddy viscosity ignores small-scale eddies and assumes the turbulent stress to be proportional
to the mean flow’s strain rate. To model compressible flows, the trace-free Smagorinsky eddy
viscosity model is implemented for the SGS stress tensor [2]. By using the compressible general-
ization of the Smagorinsky model to the SGS Reynolds’s stress tensor [6], the SGS stress tensor
6
Chapter 4.2 SGS Viscosity.
7
Mean flow.
8
Averaged total stress tensor for turbulent fluctuations in fluid momentum.
15
consists of the Smagorinsky (1963) model for the deviatoric part and the Yoshizawa (1986) pa-
rameterization for the isotropic part of the SGS stress tensor. The deviatoric part of the SGS stress,
obtained by subtracting the isotropic part of the SGS stress tensor, is responsible for the transfer
of energy between different length scales of turbulence. The isotropic part subtraction ensures the
deviatoric term to be traceless so that it does not contribute to the pressure term in the Navier-
Stokes equations:
𝜏 𝑖𝑗
𝑠𝑔𝑠 −
1
3
𝑞 2
𝛿 𝑖𝑗
= 2𝜌 ̅ 𝐶 𝑠 ∆
2
|𝑆 ̃
|( 𝑆 ̃
𝑖𝑗
−
1
3
𝑆 ̃
𝑘𝑘
𝛿 𝑖𝑗
)
(20)
where:
𝑞 2
= 2𝜌 ̅ 𝐶 𝐼 ∆
2
|𝑆 ̃
|
2
(21)
𝑆 ̃
𝑖𝑗
=
1
2
(
𝜕 𝑢 ̃
𝑖 𝜕 𝑥 𝑗 +
𝜕 𝑢 ̃
𝑗 𝜕 𝑥 𝑖 )
|𝑆 ̃
| = √2𝑆 ̃
𝑖𝑗
𝑆 ̃
𝑖𝑗
The isotropic part of the SGS Reynolds stress tensor 𝜏 𝑘𝑘
= 𝑞 2
represents the trace of the stress
tensor [2][5]. 𝑞 2
in Model (21) denotes SGS energy attributed to the compressibility of the flow,
and is parametrized using Yoshizawa’s expression [2][24] combined with a coefficient CI. Since
this term is subtracted on the left-hand side from the SGS stress tensor, the model is mathematically
“trace free.” The right-hand side term denotes the Smagorinsky model where CS is the compressi-
ble Smagorinsky coefficient. 𝑆 ̃
denotes the resolved strain-rate tensor; ∆ denotes the cutoff scale
of the LES. The equation can be re-written into Model (22).
16
𝜏 𝑖𝑗
𝑠𝑔𝑠 =
2
3
𝜌 ̅ 𝐶 𝐼 ∆
2
|𝑆 ̃
|
2
𝛿 𝑖𝑗
− 2𝜌 ̅ 𝐶 𝑠 ∆
2
|𝑆 ̃
|( 𝑆 ̃
𝑖𝑗
−
1
3
𝑆 ̃
𝑘𝑘
𝛿 𝑖𝑗
)
(22)
The subgrid heat flux 𝑞 𝑖 𝑠𝑔𝑠 uses eddy-viscosity hypothesis for the temperature (the Smagorinsky
model) [1] to relate the SGS heat flux to the resolved temperature gradient:
𝑞 𝑖 𝑠𝑔𝑠 = −𝜌 ̅ 𝐶 ℎ
∆
2
|𝑆 ̃
|
𝜕 𝐻 ̃
𝜕 𝑥 𝑘
(23)
𝐻 ̃
= 𝐶 𝑝 𝑇 ̃
For the Finite Volume discretization in which local flow information is stored in a single node of
each cell, the cutoff width is chosen to be the same order as the grid size. Model (24) gives the
cutoff scale for three dimensional cells, where Δi is the grid size in the i
th
direction.
Δ = √Δ
1
Δ
2
Δ
3
3
(24)
The three constants CI, Cs and Ch are set to be 0.12. The SGS Prandtl number is set to be 0.30.
3. 3. Immersed Boundary Method and Numerical Method Brief
The IBM is an approach for fluid structure interaction, in which the fluid mesh does not have
to conform to complex geometric structure but immerse into the regular fluid domain [20]. This
method builds up two coordinate systems in the governing equations: Eulerian x coordinate for
fluid and Lagrangian coordinate ξ for structure boundary. These two separated Individual systems
17
enable the body mesh to be embedded into the fluid mesh. Eulerian coordinate tracks fluid prop-
erties as functions to space and time and Lagrangian coordinate tracks the structure as functions
of time. A compressible IBM method is introduced by [10], assuming the capsule to have no-slip
9
and adiabatic
10
rigid surface. Zero structure deformation is also assumed in this case.
Figure 3. 2 Plane view of the capsule shoulder immersing into the fluid cell
This thesis only gives briefs of the IBM, whereas it is a vital part of the fluid solver imple-
mented for the simulation because a method is required to realize and enforce the boundary con-
ditions. A criterion must be implemented to realize the IBM coupling cells. The coordinates of the
fluid cells do not necessarily underlie the body cells that offsets between two nodal values must
be treated through mathematical methods. Therefore, coupling and interpolation are operated for
transferring data between two types of meshes. The operations smear the boundary forces on to
the body cells by interpolating forces onto the Lagrangian points [11]. The force fi in Model (13)
is implicitly calculated by a vector transform relation of normal and shear direction between two
coordinate systems. Together working with the assumptions made for the body boundary, the Na-
vier-Stokes equations can solve the boundary flow by the IBM.
Shock waves can be captured in CFD by utilizing the characteristics of jumping conditions
through an upwind method which only takes previous cells for computation. The Finite Volume
9
Zero-velocity.
10
No heat transfer across the boundary.
18
Method is implemented for spatial discretization. While keeping the time terms on the left-hand
side, the spatial terms in the governing equations can be grouped into dispersion terms including
convection, diffusion, and reaction on the right-hand side. The solver uses cell centered FVM,
solutions are evaluated at each cell’s center and fluxes are evaluated at the cell faces for ensuring
conservation. A semi-discretized system is obtained:
𝑑 U
𝜕𝑡
=
1
𝑉 ∑ 𝐅 𝑖 ,𝑘 𝑛 𝑖 𝑆 𝑘 𝑘 −𝑓𝑎𝑐𝑒𝑠 +
1
𝑉 ∑ F
𝑖 ,𝑘 𝑣 𝑛 𝑖 𝑆 𝑘 𝑘 −𝑓𝑎𝑐𝑒𝑠 + 𝑓
(25)
where 5
th
order upwind method is applied to spatial discretization, and 3
rd
order Strong Stability
Preserving (SSP) explicit Runge-Kutta method is applied for time discretization. The Courant
number (CFL) is set as 0.5 for stability that enforces the ratio of a flow particle traveled in the grid
size during a time step, determining the maximum allowable time marching size. Multiple 2
nd
order
flux evaluation methods are applied at cell faces. Distinguishment by velocity gradient for viscous
flux calculation is used for accurate evaluations on small-scale flows, and inviscid fluxes are cal-
culated elsewhere [1][10].
3. 4. Fluid Domain and Boundary Conditions
For high Reynolds number flows, one should consider the allocation of computing resources
on grid resolution, which can be achieved by means of graded meshes [17]. SAMRAI developed
by LLNL [13] is an Adaptive Mesh Refinement (AMR) tool capable of improving the accuracy of
numerical solutions while controlling the grid resolution. AMR addresses refinement in regions of
interest while maintaining a coarser mesh in the other areas of a domain: it automatically allocates
multiple layers of dynamic cell patches through time. This allows for an improved accuracy of
19
fluid-solid interaction (the IBM) while reducing the computational cost of the simulation and re-
solving flow features. With controllable refresh rate, the AMR adapts refinements on important
flow features such as boundary layers, shock waves and wakes in the fluid flow by filtering pres-
sure gradient and velocity gradient.
Figure 3. 3 Computational domain and a slice view of mesh to the x-y plane
Assuming the capsule being placed in an x-y-z field where its longitudinal axis coincides with
the fluid field’s x-axis pointing at the negative direction to an inflow boundary, Dirichlet boundary
conditions are imposed for both the fluid boundaries and structure boundaries.
Table 2. Grid dimensions of refinement levels
Level Grid dimension (m)
0 0.96 × 0.96 × 0.96
1 0.48 × 0.48 × 0.48
2 0.24 × 0.24 × 0.24
3 0.12 × 0.12 × 0.12
4 0.06 × 0.06 × 0.06
5 0.03 × 0.03 × 0.03
The IBM divides mesh into 2-dimensional triangular cells and 3-dimensional cells for the fluid.
Cell dimensions follow a rule of thumb to have an aspect ratio as close to 1 : 1 : 1 as possible for
20
best computing the approximated solutions. The capsule model is meshed using a 2-dimensional
close-equilateral triangular surface mesh with a mean area of 7.34 × 10
-4
m
2
and is embedded in a
76.8 × 38.4 × 38.4 m fluid field which is discretized into 80 × 40 × 40 grids. The resulting mesh
consisting of 128,000 cubic cells with a side length of 0.96 meters constructs the basic mesh for
the fluid field, where SAMRAI configures dynamic cell patches at complex flow structures de-
pending on pre-set refinement level. The grid size for this study is shown in Table 2: each level is
refined by a factor of 0.5 in all three dimensions, resulting in each cubic cell being subdivided into
eight finer cubic cells when necessary.
Figure 3. 4 (a) AMR cells pack (b) AMR patches at the wake
Figure 3.4 (a) gives an example of three refinement levels. Figure 3.4 (b) is a second example
showing how SAMRAI matches the cells with a turbulent flow and a recompression shock. Several
levels of cubic refinement patches are placed in assorted sizes.
Figure 3. 5 Slice view of AMR patches at a shock wave
21
Figure 3. 5 demonstrates SAMRAI’s gradient detector. The colored line is a slice view of a partial
shock wave, which is captured by high pressure gradient values stored in the cells
11
. A fluid solver
can detect refinement sections by much higher velocity gradients and pressure gradients at dis-
turbed regions as mentioned in Chapter 3. 1. The top left part of the shock wave in Figure 3. 5
has better resolution because the AMR detects higher values at this sector and updates finer cell
patches. Hence, the gradient detector is used for controlling the refinement for the flow model.
Assuming a ballistic trajectory with low ballistic coefficient β ≈ 4788 N/m
2
for the capsule
reentry, the freestream air properties are chosen at 20,000 m altitude from sea level according to
the reentry M vs. h data [22]
12
. Note that β = W/(CDAref), where W denotes the capsule weight.
Table 3. US standard atmosphere property at h = 20,000 m
ρ (kg/m
2
) T (K) P (Pa) µ (N· s/m
2
)
0.089 216.65 5229 1.42× 10
-5
The fluid domain is built by velocity-based inflow boundary at freestream direction and pressure-
based boundaries for outflow domains. Table 3 lists necessary freestream parameters according to
the 1976 U.S. standard atmosphere model [19]. The freestream speed of sound is 295.1 m/s, the
freestream velocity U∞ at the inflow is 737.72 m/s, and the thermal conductivity is 0.020 W/m· K.
The specific heat Cp = 1004.46 J/kg· K and the Prandtl number Pr = 0.7 are constant throughout
the field. The capsule is assumed to have a no-slip and adiabatic surface boundary as mentioned.
The flow solver starts with homogeneous initial conditions using values in Table 3 except U = 0
m/s at t = 0 s. The inflow velocity at the boundary is accelerated to U∞ = 737.72 m/s from t = 0 s
to t = 0.1 s for attaining faster iterations of appropriate solution fields.
11
Darker regions exhibit higher gradient values.
12
4788 N/m
2
≈ 100 lb f/ft
2
.
22
Chapter 4: Analysis of Aerodynamic Parameters
Figure 4. 1 Closeup of a slice view of the x-z plane Mach number contour and AMR refinement
through the center of the computational fluid domain
This chapter gives a comprehensive analysis for force results that aerodynamic parameters are
collected and analyzed at various AOAs. Figure 4.1 displays a slice view of the Mach contour by
Superfine solutions, depicting the fully accelerated and propagated flow at α = 0° . The mesh re-
finement is focused on the capsule vicinity, shock wave, and wake region while the non-disturbed
area retains coarse meshes as anticipated.
4. 1. Computation Error
To gain an insight of the accuracy of the results, the flow model for 0° AOA is solved by three
mesh grades categorized as Coarse, Fine, and Superfine by different numbers of refinement levels.
As previously mentioned in Chapter 3. 4 in Table 2, size factor 0.5 is applied for mesh refinement
with each ascend level while level 0 cell size stays the same. Starting from level 0, the mesh grades
23
have level 3, 4, and 5 respectively. The first data column in Table 4 gives average triangular areas
for the surface mesh on the capsule; the second data column is the fluid’s side length cells of the
highest refinement level immersed into the solid mesh, and the cube’s face areas are calculated
accordingly.
Table 4. Mesh Setup of the Fluid and Solid
Capsule Avg. mesh area (m
2
) Finest cell length d(m) Cell face area (m
2
)
Coarse 9.9× 10
-3
0.12 14.4× 10
-3
Fine 2.6× 10
-3
0.06 3.6× 10
-3
Superfine 7.3× 10
-4
0.03 9.0× 10
-4
The drag force results for three mesh qualities are plotted in Figure 4.2. The flow accelerates
from Mach 0 to 2.5 (0 m/s to 737.72 m/s) at 0 < t ≤ 0.1 s and subsequently remains constant with
the force results significantly increasing during acceleration.
Figure 4. 2 Drag results by three levels of mesh quality at α = 0°
The result groups contain several types of errors in which discretization error and truncation error
play a dominant role. The discretization error, also known as numerical error, is caused by numer-
ical approximation methods applied during spatial and temporal discretization onto the continuous
24
PDE equation. Note that the numerical method of the IBM also causes error. The truncation error
is the difference between the PDE and the finite equation. For the LES, the truncation error exists
due to the filtering operation. Through analyzing Figure 4. 2, drag force differences can be found
as a reflection of mesh quality. At 0.1 < t ≤ 0.2 s, it shows a tendency of decreasing deviation as
mesh quality improves, indicating that the discretization error would be eliminated with mesh size
decreasing to zero. Besides residual checks conducted for spatial convergence at iterations for each
time step, the high consistency of drag force results also validates the temporal convergence of the
fluid solver. The error in the numerical solution of α = 0° can be quantified through Richardson
extrapolation for three level groups. Although the extrapolation method does not completely elim-
inate the error in the solutions, it provides estimations that tolerably a set of results named Exact
supporting error analysis is obtained, see Figure 4. 2 and Appendix 1.
Table 5. Maximum solution error at 0.1 < t ≤ 0.2 for α = 0°
F (N) Pmax (Pa) Tmax (K) ρmax (kg/m
3
)
3.28% 1.09% 2.64% 0.89%
4. 2. Numerical Analysis on Aerodynamic Parameters
Solution fields are acquired by Superfine mesh with each having approximately 47 million
cells so that the errors of the computed results for 2° , 5° and 10° AOAs are expected to be simi-
lar. To evaluate the aerodynamic performance at these AOAs, the results by force component are
obtained. The upper half error bars were evaluated based on the deviation between the results of
Superfine and Fine (3.07%), while the lower half error bars were based on the deviation between
the results of Superfine and Exact (3.28%)
13
.
13
Both are evaluated by the maximum values at 0.1 < t ≤ 0.2 s at α = 0° .
25
(a) α = 0°
(b) α = 2°
Figure 4. 3 Drag results at M = 2.5 (a) & (b)
26
(c) α = 5°
(d) α = 10°
Figure 4. 3 Drag results at M = 2.5 (c) & (d)
27
(a) α = 0°
(b) α = 2°
Figure 4. 4 Lift (Fy) results at M = 2.5 (a) & (b)
28
(c) α = 5°
(d) α = 10°
Figure 4. 4 Lift (Fy) results at M = 2.5 (c) & (d)
29
(a) α = 0°
(b) α = 2°
Figure 4. 5 Fz results at M = 2.5 (a) & (b)
30
(b) α = 5°
(d) α = 10°
Figure 4. 5 Fz results at M = 2.5 (c) & (d)
31
A flow passing over a symmetric geometry theoretically produces a symmetric flow distribu-
tion, but the unsteady state causes fluctuations. Drag (D = –Fz) and lift (Fy) results in Figure 4. 3
and Figure 4. 4 show that the amplitude of the oscillation decreases as time proceeds and ap-
proaches a dampened state. This phenomenon is only valid for non-zero forces, whereas small
fluctuations are dominant at planes with symmetric geometry: vertical lift and lateral forces pre-
sented in Figure 4. 4 (a) and Figure 4. 5 (a) − (d) has undamped oscillation throughout the time
scale with apparent amplitudes. The frequency of the axial-force oscillation is considerably high
that may cause vibration to the capsule. Additionally, it is noteworthy that because of the SGS
modeling assumptions, some high frequency SGS information is cut off from the solutions.
Table 6. Average force results at 0.2 < t ≤ 0.25 s
α (° ) Drag (× 10
3
N) Lift (× 10
3
N) Fz (N)
0 416.00 0.020 -10.38
2 415.65 10.66 57.08
5 413.85 26.30 -30.75
10 407.42 51.63 -100.17
The average results of forces are acquired in Table 6. While the lift increases as anticipated, de-
creasing in drag is determined along with increasing AOA from 0 − 10° . A 10° AOA generates
51.63 k
N lift being comparatively low because a high-pressure region in front of the capsule base
produces an integrated drag force around 407.42 k
N. The drag load is reduced by 5.58 kN (2.06%)
with respect to 0° AOA.
The aerodynamic parameters L/D, CL and CD are tabulated in Table 7 using Model (1) − (3)
and average force results obtained in Table 6. Typically, 0.2 < t ≤ 0.25 s given by damped Super-
fine results is of interest for calculating average values for L/D, CL, and CD. The L/D demonstrates
32
a poor aerodynamic lift performance of the capsule geometry, though the objective of a reentry
mission is deceleration that requires a large drag force exerting on the capsule.
Table 7. Aerodynamic parameter results at at 0.2 < t ≤ 0.25 s
α (° ) L/D CL CD
0 0.000 0.000 1.54
2 0.026 0.039 1.53
5 0.064 0.097 1.53
10 0.127 0.191 1.50
The parameters are also plotted in Figure 4. 6, demonstrating a linear relationship for L/D vs. α
and CL vs. α at 0° ≤ α ≤ 10° . See Appendix 3 for unsteady L/D solutions. Small variations in CD
are observed as AOA increases, with a slight decrease in its value demonstrating a typical drag
performance of blunt geometry. Despite the results obtained for four AOAs, the discrete figures
only provide a coarse indication of the capsule’s aerodynamic performance, where the maximum
L/D = 0.127 is determined at α = 10° with CL = 0.191 and CD = 1.50. As a reference, Orion Crew
Module has L/D ≈ 0.17, CL ≈ 0.25 and CD ≈ 1.45 at Mach 2.5 [23].
(a) Lift-drag ratio
33
(b) Lift coefficient (c) Drag coefficient
Figure 4. 6 Aerodynamic parameters
The reentry aerodynamic performance is a sophisticated topic for capsule geometry design as
the mission evolves a wide range of flow conditions from the ceiling of the earth’s atmosphere to
the ground, involving drastic change in the surrounding flow. The L/D is an extraordinarily com-
plex function of aerodynamics and center of gravity (CG), with the parameters being probabilistic,
yet the mass distribution is undefined for the capsule. The aerodynamic lift force (L) and drag force
(D) are nonlinear functions of the position, velocity, and attitude [16]. While this study demon-
strates a linear relation between L/D and α at a fixed altitude, CL and CD are also affected by the
Reynolds number Re (U∞, μ(T∞))
14
and the Mach number M (U∞ , T∞) with relation to freestream
velocity and temperature that in turn affects the flow properties and the boundary layer behavior
around the capsule. In fact, the Mach number and the viscosity of the flow model is assumed only
as a function of a fixed freestream temperature, whereas T = f(h) when broad range of altitude is
taken into consideration and significantly affects the L/D.
14
Aerodynamic force contributed by the viscous force.
34
Chapter 5: Flow Visualization and Analysis
Figure 5. 1 Schematic diagram of the flow compressibility at center x-y plane at α = 0°
Pseudo-color diagrams are used for investigating supersonic flow features for the capsule ge-
ometry by the 0° AOA solution field
15
at t = 0.248 s unless specified otherwise. This high Reyn-
old’s number flow is subject to viscous interactions associating with strong flow separations and
strong turbulences. Figure 5. 1 is a schematic diagram by density depicting compressed regions
with warm colors and expanded regions with cool colors. Several regions are assigned with char-
acters for indication. The supersonic flow features discontinuity across a leading shock and strong
perturbations at wake. A highly compressed zone is formed between the leading shock and the
capsule base that most of the kinetic energy converts into the heat energy. A long wake behind the
capsule is designated by Zone (a) − (c) referring to a recirculation
16
zone, a transition zone and a
turbulent zone. Zone (d) is a bypass where the expanded air accelerates to velocity slightly higher
than the freestream flow encircling the wake. Zone (e) is a pressure wave caused by the leading
shock bouncing at the fluid boundaries.
15
30000
th
time step.
16
Flow trapped in swirling motions.
35
Figure 5. 2 Three-dimensional perspective of ∇Ux contour
Figure 5. 3 Slice view of center x-z plane by the Mach number
The velocity gradient contour Figure 5. 2 visualizes the leading shock
17
and the wake: abrupt
increases of rate change of velocity towards the downstream direction exist at leading shock, shear
layers and turbulent structures. The leading shock is a bow shock approximately 1.0 m ahead from
the capsule base. Despite the geometry of the leading shocks showing minor variations as the AOA
increases, the compression at the bottom half of the base becomes stronger, and the local shock
17
∇U x ≈ 10000 s
-1
ahead of the capsule and decreases along the shock wave.
36
distances are closer to the capsule, see Table 8. Upon observing the solution fields at 0.1 < t ≤ 0.25
s, it is found that the leading shock exhibits fluctuations, generating significant unsteady flow var-
iables with oscillations at the highly compressed zone ahead of the capsule base and the wake
nearby. Figure 5. 3 is a sliced view of Mach number characterizing flow compressibility, see also
Figure 5. 7 (a). Multiple Prandtl-Meyer expansion waves are detected at the capsule shoulder. The
flow is compressed by about 3.8 times, the pressure by about 9 times, and the temperature by about
2.3 times, see Appendix 2. Although the pressure wave is caught lowering the minimum temper-
ature at bypass flow
18
, its impact does not harm upstream flow behavior.
Table 8. Leading shock distance to the capsule base
α (° ) Minimum distance (m)
0 1.00
2 1.00
5 0.96
10 0.90
At α = 0°, the flow around the capsule is axisymmetric with equal pressure distribution. The
non-axisymmetric flows at non-zero AOAs result in pressure difference between the upper-half
and lower-half capsule surface that generates lift and attenuated drag. The attenuated drag forces
at higher AOAs in Chapter 4. 2 are due to the flight attitude with part of the pressure force con-
tributing to the down force (−Fy). The pressure distribution on the lower-half capsule surface be-
comes more dominant and stable than the upper-half capsule surface with increasing AOAs due to
the suppression of fluctuating space inside the wake, thus reducing the force oscillation amplitudes.
18
T bypass ≈ 210 K.
37
(a) α = 0°
(b) α = 10°
Figure 5. 4 Slice view of center x-y plane by the pressure (kPa)
38
Figure 5. 5 Slice view of center x-y plane by the Mach number with Mmin = 2.0
Figure 5. 6 Shear layers captured around the capsule by ∇Ux with ∇Ux max = 1500 s
-1
Table 9. Flow properties at the resolved layers
U (m/s) ∇Ux (s
-1
)
Unsteady shear layers × ×
Turbulent shear layer 200-600 ×
Outer turbulent layer 750 ×
Bypass layer 1 700 12.5
Bypass layer 2 740 6.3
“×” = Unpredictable / significant variation.
Values are roughly estimated.
39
The flow detaches at the capsule shoulder, forming a recirculation zone. A strong detached
boundary layer, namely a shear layer with strong velocity gradient
19
, deflects most of the flow to
the bypass. This layer
20
is defined as the wake boundary in this thesis, consisting of three sublayers
at the resolved scale. The wake boundary layer exhibits significant velocity gradient change as it
propagates. For the high Reynolds number flow, these layers are completely turbulent shear layers
but remain as non-visualizable SGS features due to insufficient resolution. A thin boundary layer
adhering to the structure surface is also an SGS feature not captured by the fluid domain that
eventually separates as well. Figure 5. 6 shows unsteady shear layers
21
forming ahead of the cap-
sule base. One can also find unsteady small shear layers “peel off” from the capsule surface along
the wake path and disturb the flow (U ≈ 50~120 m/s). These “peeling off” motions observed are
computation results caused by the IBM used in the fluid solver, though they do not cause evident
perturbation to the pressure distribution or forces. Table 9 summarizes all resolved layer features
around the wake, showing an increasing tendency of the velocity away from the longitudinal axis.
There are two weak laminar layers between the wake and the bypass flow not clearly recognized
in Figure 5. 3. By filtering with minimum Mach value in Figure 5. 5, one can distinguish two
layers in the green and yellow regions flowing at about Mach 2.3 and Mach 2.5 disturbed by the
pressure wave.
The recirculation flow transforms into fully developed turbulent flow enveloped by an outer
turbulent layer. A recompression shock appears along the time to straighten the bypass flow. The
shock confronts the wake boundary layer, where this turbulent shear layer acts like ramps. Com-
pared to the leading shock, the recompression shock is weaker with less abrupt changes in the flow.
19
Equivalently strong viscous shear.
20
Named as the wake boundary layer from this point.
21
Relatively significant local velocity gradients.
40
(a) ρ (kg/m
3
)
(b) T (K)
Figure 5. 7 Slice view at the capsule through center x-y plane of the compressibility
Figure 5. 8 Wake features decomposition of center x-z plane Mach field
41
Figure 5. 7 presents sliced views of density and temperature at Wake Zone (a) and vicinity,
where the density distribution is consistent with Figure 5. 1. The expanded flow in the recircula-
tion zone is slow, heated, and disordered, with the compressed hot air ahead of the capsule to
observed flowing into the wake with strong expansion that envelopes the capsule at approximately
480 K. Local recompression and expansion with temperature change are determined varying with
changing AOAs.
The recirculation flow propagates for some distance and soon develops into large turbulent
structures wrapped by the turbulent shear layer, which can be explained by several phenomena: (1)
Chaotic flow motions are produced in the recirculation zone. (2) The wake boundary layer con-
taining strong SGS eddy structures contracts towards the longitudinal axis. (3) The wake re-accel-
erates. It is determined that, by tracing the unsteady models over time, the flow separation gener-
ates produces periodic waves transferring downstream along the wake boundary layer as pointed
out by white arrows in Figure 5. 8. The perturbation is subsequently amplified during propagation.
Phenomenon (2) and Phenomenon (3) are caused by fluid mixing motion. A reattachment zone
forms downstream of the recirculation zone that the wake experiences acceleration because of the
fast-moving bypass flow. The mixing simultaneously reattaches slow-moving eddies and vigor-
ously enhancing instability. With eddies inside growing and interacting, the coherent boundary
eventually collapses and large supersonic turbulence forms.
Figure 5. 9 (a) – (d) presents U, ρ, P, T and E fields at the turbulent wake in sequence. The
turbulent structures have lower velocity, density and pressure, and are gradually decaying due to
the mixing with the bypass flow. The temperature in the wake region tends to decrease with accel-
erating turbulence as the internal energy transforms into kinetic energy that causes the flow inside
the wake to cool.
42
(a) Velocity (m/s)
(b) Density (kg/m
3
)
(c) Pressure (kPa)
43
(d) Temperature (K)
(e) Total energy (kJ)
Figure 5. 9 Slice view of flow variables at the turbulent wake through center x-y plane
44
(a) α = 0°
(b) α = 2°
45
(c) α = 5°
(d) α = 10°
Figure 5. 10 Closeup slice view at center x-y plane of AOAs by the Mach number for α =
0° , α = 2° , α = 5° and α = 10° at t = 0.248 s
46
For 0° , 2° and 5° AOA models, Ká rmá n vortex street exists at the turbulent wake where repeat-
ing vortex patterns are shed from the wake. The unsteady flow produces asymmetric vortices in
the wake detaching from the main wake structure periodically. The force results in Chapter 4. 2
reveal that lift fluctuations are evidently suppressed with lower amplitudes at α = 10°. Meanwhile,
non-zero AOAs with asymmetric flow paths along the capsule surface cause the wake to deviate
from the longitudinal axis. By comparing these results with Figure 5. 10, it indicates a relation
between upstream fluctuations and downstream shedding, and a possible critical AOA leading to
suppression of the shedding effect.
Figure 5. 11 Closeup view of central x-y plane at the capsule shoulder by unit velocity vector
High Reynolds number flow is a complicated topic as the LES may not reflect the exact turbu-
lent performance of the model. The assumptions made by the SGS modeling result in loss of some
small information, especially in the wake region with plenty of complex SGS features that can
cause possible deviations in flow propagation, meaning the fluid solver may compute enhanced or
suppressed flow disturbances such as the vortex shedding effect determined in this simulation.
Although the SGS model directly affects the resolved scale, the resolved fields do not present
fluid motion in detail since the SGS features are non-visualizable in the solution. For example, it
is difficult to verify if flow exchanges across the detached wake boundary layer because this
47
turbulent shear layer is resolved as coherent structures
22
surrounding the wake; the motion of the
thin adhering boundary layer also remains undetermined. The combination of the LES and the
IBM produces some inappropriate flow paths that the fluid-structure treatment by interpolation
does not strictly enforce zero velocity at the immersed cells, resulting in inaccurate integration of
streamlines. On the other hand, it shows the advantage of the LES that successfully accompanies
the complex flow beyond resolution for approximating fluid motions of high Reynolds numbers.
22
The resolved layers with filtered flow property values in Table 9.
48
Chapter 6: Conclusion
Precisely describing turbulent flow by mathematics is of huge complexity as it involves mul-
tiple factors. The Large-Eddy Simulation provides a practical method using sustainable computa-
tion resources for capturing key information from a resolved flow field, which scopes in the aero-
dynamics study for the capsule. This study concludes model stability within the time step accord-
ing to the damped solutions attained by the fluid solver.
Flow metrics are well investigated. The resolved models closely simulate the reentry super-
sonic flow and predict aerodynamic parameters for the prototype capsule well, revealing high load
pressure and heat conditions in which the intense pressure distribution produces over 400 kN of
drag and the heat generation forms an approximately 480 K hot layer surrounding the capsule
surface. The strong drag loads caused by pressure distribution require robust structure design to
overcome deformation. In contrast, the capsule has weak lift performance with the reason ad-
dressed to the low pressure around the capsule body due to its geometry. A leading bow shock
exists approximately 1.0 m ahead from the capsule base in each model with no significant geom-
etry variation at 0° , 2° , 5° , or 10° AOAs. The wake, however, is affected by the capsule attitude
with respect to the AOAs studied along with the strong turbulent shear layer. Recompression shock
and large turbulent structures are determined downstream as well. The oscillations in the solution
fields are caused by the unsteady leading shock and the strong flow separation with massive eddies
produced inside the wake region. The SGS model plays a successful role in computing flow re-
gions with high complexity, indicating that the LES is a highly promising method for simulating
supersonic flows.
49
References
[1] Hang Yu, Carlos Pantano, Fehmi Cirak, “Large-Eddy Simulation of Flow Over Deformable
Parachutes using Immersed Boundary and Adaptive Mesh,” AIAA Scitech 2019 Forum, 2019-
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[2] Moin, P., Squires, K., Cabot, W., and Lee, S., “A Dynamic Subgrid-Scale Model for Com-
pressible Turbulence and Scalar Transport,” Physics of Fluids A, Vol. 3, 1991, pp. 2746–2757.
[3] R. M. J. Kramer, F. Cirak, and C. Pantano, “Fluid–Structure Interaction Simulations of a Ten-
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[5] T. Sayadi AND P. Moin, “Predicting Natural Transition using Large-Eddy Simulation,” Center
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tion of Compressible Turbulent Flows,” ICASE Report 90-76.
[7] Vreman, B., Geurts, B., Kuerten, H. (1995), “A Priori Tests of Large-Eddy Simulation of the
Compressible Plane Mixing Layer,” J. Eng. Math. 29, 299–327
[8] Smits, A. J., Dussauge, J. P. (2006): Turbulent Shear Layers in Supersonic Flows, 2nd edn.,
Springer, Berlin
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S., “The immersed boundary method,” Acta Numerica, Vol. 11, 2002.
[10] HangYu, Carlos Pantano, “An immersed boundary method with implicit body force for
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[11] Kunihiko Taira, Tim Colonius, “The immersed boundary method: A projection approach,”
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[12] Wang Ronga, Chen Bing-yana, Zhang Hong-juna, Zhou Wei-jianga, Bai Penga, Yang
Yun-juna, “Aerodynamic Design Optimization of a Kind of Reentry Capsule Based on CFD
and Multi-objective Genetic Algorithm,” 2014 Asia-Pacific International Symposium on Aer-
ospace Technology, APISAT2014.
[13] “SAMRAI Project Site: Center for Applied Scientific Computing,” Lawrence Livermore
Laboratory, Livermore, CA, 2012. URL http://www.llnl.gov/CASC/SAMRAI.
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[14] NASA – “Orion Crew Exploration Vehicle (Press release)". NASA. February 7, 2009.
URL https://www.nasa.gov/pdf/156298main_orion_handout.pdf
[15] Andrew Davidhazy, “Supersonic Bullet Shadowgram in Air Showing Bow Shock and Tur-
bulence,” The Physics Teacher 52, 402 (2014)
[16] Jeremy R. Rea, “Orion Entry Performance-Based Center-of-Gravity Box,” AIAA Journal,
AIAA 2010-8061.
[17] H. K. Versteeg and W. Malalasekera - An Introduction to Computational Fluid Dynamics:
The Finite Volume Method 2
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Edition - ISBN: 978-0-13-127498-3.
[18] John D. Anderson Jr.- Introduction to Flight - 8th Edition, ISBN 978-0-07-802767-3.
[19] National Oceanic and Atmospheric Administration, National Aeronautics and Space Ad-
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[20] Jiyuan Tu, Guan-Heng Yeoh and Chaoqun Liu - Computational Fluid Dynamics: A Prac-
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boundary method for the numerical simulation of compressible flows,” J. Comput. Phys. 374
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[22] John C. Adams Jr., “Atmospheric Re-Entry,” Arnold Engineering Development Center,
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Entry.pdf
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51
Appendix 1: Results and Errors ( α = 0° )
(a) Drag D
(b) Global ρmax
(c) Global Pmax
52
(d) Global Tmax
Note: errors are evaluated by the Superfine results
53
Appendix 2: Global Maximum and Minimum Flow Properties at M = 2.5
(a). Averaged global pressure results at 0.2 < t ≤ 0.25 s
α (°) Pmax (kPa) Pmin (Pa) Pmax/P∞ Pmin/P∞
0 47.05 3872 9.00 0.74
2 46.96 3819 8.98 0.73
5 47.16 3915 9.02 0.75
10 46.84 3821 8.96 0.73
(b). Averaged global density results at 0.2 < t ≤ 0.25 s
α (°) ρmax (kg/m
3
) ρmin (kg/m
3
) ρmax/ρ∞ ρmin/ρ∞
0 0.34 0.029 3.78 0.32
2 0.34 0.028 3.77 0.32
5 0.34 0.028 3.78 0.31
10 0.34 0.027 3.77 0.31
(c). Averaged global temperature results at 0.2 < t ≤ 0.25 s
α (°) Tmax (K) Tmin (K) Tmax/T∞ Tmin/T∞
0 508.08 205.64 2.35 0.95
2 505.99 205.35 2.34 0.95
5 507.78 205.51 2.34 0.95
10 507.83 205.31 2.34 0.95
54
Appendix 3: Unsteady L/D Results at M = 2.5
(a) 0 AOA L/D
(b) 2 AOA L/D
55
(c) 5 AOA L/D
(d) 10 AOA L/D
56
Appendix 4 (a): Slice View of the Mach Number at M = 2.5
(a) α = 0°
(b) α = 2°
57
(c) α = 5°
(d) α = 10°
58
Appendix 4 (b): Slice View of the Total Energy (kJ) at at M = 2.5
(e) α = 0°
(f) α = 2°
59
(g) α = 5°
(h) α = 10°
60
Appendix 5: U.S. Standard Atmosphere Air Properties - SI Units
Abstract (if available)
Abstract
This study presents a flow simulation of a 3.89 m diameter capsule at Mach 2.5 dedicated to investigating various aspects of its flow metrics. The high Reynolds number flow exceeding 10^7 results in strong separation of the boundary layer producing a complicated wake with high dissipation energy. With proper assumptions made, the study utilizes Large-Eddy Simulation (LES) using a compressible subgrid-scale (SGS) model coupled with the Immersed Boundary Method (IBM) for cheaper computation approach simulating flight attitudes at 0°, 2°, 5° and 10° angle of attack. Air properties are presumed at 20,000 m altitude for the freestream flow along with proper boundary conditions and initializations for the fluid field. The solution fields obtained provide an insight into the compressible flow behavior over the capsule that leading shock waves are success-fully captured for each model. Recompression shocks and vortex shedding effects are also caught along the strong shear layers produced by the capsule geometry. Averaged numerical results indicate minor compressibility change within the AOAs being studied, though significant differences of wake behaviors are determined. Increasing values from 0° − 10° AOAs changes pressure force distribution with over 400 kN drag forces and lift-drag ratios ranging from 0 − 0.127.
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Ma, Junhao
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Supersonic flow study of a capsule geometry using large-eddy simulation
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Master of Science
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Aerospace Engineering
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2023-05
Publication Date
05/03/2023
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