I Framework
1 Context and motivation 1-1
1.1 Space exploration . . . 1-1 1.2 Atmospheric entry . . . 1-2 1.3 TPS ablation . . . 1-4 1.4 Boundary-layer transition . . . 1-5 1.5 Thesis scope . . . 1-7 References . . . 1-10 2 Thermophysical framework 2-1
2.1 Thermodynamic flow assumptions . . . 2-1
2.2 Gas mixtures and reactions . . . 2-3
2.3 Flow equations . . . 2-5
2.3.1 Thermo-chemical non-equilibrium (TCNE) . . . 2-7
2.3.2 Chemical non-equilibrium (CNE) yet in thermal equilibrium2-10
2.3.3 Local thermo-chemical equilibrium with elemental
demix-ing (LTEED) . . . 2-10
2.3.4 Local thermo-chemical equilibrium with constant
elemen-tal composition (LTE) . . . 2-13
2.3.5 Thermal non-equilibrium yet chemically frozen (TNE) . . 2-14
2.3.6 Thermo-chemically frozen gas (TCFG) . . . 2-15
2.3.7 Thermally perfect and chemically frozen gas (TPG) . . . . 2-15
2.3.8 Calorically perfect gas (CPG) . . . 2-16
References . . . 2-17
3 Stability and transition 3-1
3.1 Linear theories . . . 3-4
3.1.1 LDNS . . . 3-4
3.1.2 Tri-global stability . . . 3-5
3.1.3 Bi-global stability or 2D-LST . . . 3-5
3.1.4 Linear stability theory . . . 3-7
3.1.5 Linear parabolized stability . . . 3-8
3.3 Non-linear theories . . . 3-10
3.4 Adjoint theories . . . 3-11
3.5 Physical implications of the choice of the wave parameters . . . . 3-11
3.6 Instability mechanisms . . . 3-12
References . . . 3-14
4 Gas properties 4-1
4.1 Thermal properties . . . 4-5
4.2 Transport properties . . . 4-8
4.2.1 Viscosity and thermal conductivity . . . 4-8
4.2.1.1 Blottner-Eucken-Wilke model . . . 4-9
4.2.1.2 Gupta-Wilke model . . . 4-9
4.2.1.3 Sutherland-Eucken-Wilke and
Sutherland-Wilke models . . . 4-10
4.2.1.4 Chapman-Enskog model . . . 4-10
4.2.1.5 Brokaw and Yos models . . . 4-11
4.2.2 Diffusion models . . . 4-12
4.2.3 Collisional models . . . 4-16
4.3 Chemical properties . . . 4-21
4.4 Gas-surface interaction model . . . 4-25
4.5 Energy exchange terms . . . 4-28
4.5.1 Vibrational source term . . . 4-29
4.5.2 Electronic source term . . . 4-30
4.5.3 Electron source term . . . 4-30
4.6 Thermodynamic derivatives . . . 4-32
4.6.1 Thermodynamic derivatives of the Chapman-Enskog
ma-trix system . . . 4-34
4.6.2 Thermodynamic derivatives of the Stefan-Maxwell matrix
system . . . 4-35
References . . . 4-36
5 Laminar base flow 5-1
5.1 Boundary-layer equations and their self-similar solutions . . . 5-1
5.1.1 CNE boundary-layer equations . . . 5-5
5.1.2 LTEED boundary-layer equations . . . 5-6
5.1.3 LTE boundary-layer equations . . . 5-8
5.1.4 TNE boundary-layer equations . . . 5-8
5.1.5 TCFG boundary-layer equations . . . 5-9
5.1.6 TPG and CPG boundary-layer equations . . . 5-9
5.1.7 Axysimmetric boundary layers – the Probstein-Elliot
transformation . . . 5-10
5.2 Base-flow and stability hypotheses . . . 5-12
5.3 Base-flow inaccuracy propagation to stability predictions . . . 5-13
5.3.1 Base-flow grid convergence . . . 5-14
5.3.3 Stability grid convergence . . . 5-15
5.3.4 Base-flow-induced errors . . . 5-17
5.3.5 The importance of performing joint base-flow-stability
grid convergence studies . . . 5-20
5.3.6 Conclusions . . . 5-20
References . . . 5-23
6 Boundary conditions 6-1
6.1 Wall boundary . . . 6-1
6.1.1 Blowing boundary condition . . . 6-2
6.1.1.1 Porous stability boundary condition (PSBC) . . 6-3
6.1.1.2 Ablation-mimicking continuously-blowing
sta-bility boundary condition (AMSBC) . . . 6-5
6.1.2 Thermal boundary conditions . . . 6-6
6.1.3 Concentration boundary conditions . . . 6-7
6.1.4 Perturbation compatibility boundary conditions . . . 6-8
6.2 Freestream boundary . . . 6-9
6.2.1 Shock boundary condition . . . 6-11
References . . . 6-13
II Tools
7 DEKAF flow solver 7-1
7.1 Shock-jump relations . . . 7-2
7.2 Inviscid non-equilibrium 1-D solver . . . 7-5
7.3 Numerical method . . . 7-6
7.3.1 Mappings . . . 7-9
7.3.2 Initial guess . . . 7-11
7.4 Exploiting spectral accuracy through the GICM interpolation . . . 7-13
7.5 Implementation . . . 7-15
7.6 Verification . . . 7-17
7.6.1 CPG non-bleeding flat-plates . . . 7-17
7.6.2 CPG bleeding flat-plates . . . 7-17
7.6.3 Incompressible Falkner-Skan-Cooke equation . . . 7-19
7.6.4 CPG cone . . . 7-21
7.6.5 CNE Mach 10 flat plate . . . 7-22
7.6.6 CNE Oxygen mixture . . . 7-22
7.6.7 6-degree CPG, CNE, and LTE wedge . . . 7-23
7.6.8 20-degree CPG, TPG, and CNE wedge at Mach 18 . . . . 7-26
7.6.9 15-degree CNE wedge at Mach 35 . . . 7-26
7.6.10 TNE flat plate . . . 7-28
7.6.11 10-degree CPG, CNE and LTE cone at Mach 25 . . . 7-29
7.6.12 State-of-the-art properties . . . 7-32
References . . . 7-34
8 VESTA’s Automatic Derivation and Implementation Tool 8-1
8.1 Stability equations . . . 8-2
8.1.1 Non-dimensionalization convention . . . 8-3
8.1.2 Treatment of the spatial derivatives of the dependent
per-turbation quantities Q� . . . . 8-6 8.1.3 Implementation . . . 8-7 8.1.3.1 Derivation options . . . 8-9 8.1.3.2 Provisional variables . . . 8-11 8.1.3.3 In-line derivatives . . . 8-13 8.1.4 Verification . . . 8-13 8.1.4.1 CPG flow assumption . . . 8-14
8.1.4.2 Porous boundary condition . . . 8-14
8.1.4.3 LTE flow assumption . . . 8-15
8.1.4.4 Shock boundary condition . . . 8-16
8.1.4.5 CNE flow assumption for 2 species . . . 8-18
8.1.4.6 TCNE and CNE flow assumption for 5 species . 8-18
8.1.4.7 TCNE flow assumption for 5 species . . . 8-19
8.1.4.8 Orthocurvilinear coordinate system . . . 8-19
8.2 Gas properties . . . 8-21
8.3 Summary . . . 8-24
References . . . 8-25
III Results
9 Surface outgassing and transition 9-1
9.1 Experimental-numerical investigation of an outgassing cone at
Mach 6 . . . 9-2
9.1.1 Experimental setup . . . 9-2
9.1.2 Numerical setup . . . 9-4
9.1.3 Validation of the porous boundary condition . . . 9-5
9.1.4 Interpretation of the experimental results based on the LST
analysis . . . 9-7
9.1.5 Evaluation of non-parallel effects . . . 9-10
9.1.6 Conclusions . . . 9-11
9.2 Sensitivity of instabilities to the composition of the injection gas . 9-12
9.2.1 Test conditions and gas modeling . . . 9-13
9.2.2 Case 1: self-similar multi-species blowing . . . 9-15
9.2.2.1 Span-wise wavenumber sweep . . . 9-23
9.2.2.2 Transport model comparison . . . 9-23
9.2.2.3 PSBC-HSBC comparison . . . 9-26
9.2.3 Cases 2-5: non-self-similar He blowing . . . 9-26
9.2.5 Conclusions . . . 9-34
9.3 Sensitivity of instabilities to the wall boundary condition . . . 9-36
9.3.1 Baseline case – homogeneous condition . . . 9-37
9.3.2 Porous condition for case O . . . 9-38
9.3.3 Cases M-, M+, O, T-, and T+ for all boundary conditions . 9-41
9.3.4 Porous stabilizing/destabilizing maps and experimental
re-sult analysis . . . 9-45
9.3.5 Conclusions . . . 9-47
References . . . 9-49
10 High-enthalpy effects and transition 10-1
10.1 Gas modeling influence on stability predictions . . . 10-1
10.1.1 Case 1 – influence of the transport model . . . 10-2
10.1.2 Case 2 – influence of the diffusion model . . . 10-5
10.1.3 Cases 1 and 2 – influence of the collision model . . . 10-7
10.1.4 Case 3 – influence of the chemical equilibrium model . . . 10-11
10.1.5 Case 4 – influence of the chemical kinetic model . . . 10-11
10.1.6 Case 1 – effects of transport modeling on perturbation terms10-15
10.1.7 Mode shapes . . . 10-15
10.1.8 Conclusions . . . 10-15
10.2 Enhanced and weakened diffusion fluxes . . . 10-19
10.3 Flow-assumption adequacy for different earth entry flow regimes . 10-22
10.3.1 Laminar base-flow field . . . 10-23
10.3.2 LST N-factor curves . . . 10-27
10.3.3 LPSE N-factor curves . . . 10-28
10.3.4 Conclusions . . . 10-28
10.4 Ionization and dissociation . . . 10-30
10.4.1 Laminar base flow . . . 10-32
10.4.2 LST analyses . . . 10-37
10.4.2.1 Consistent base-flow and perturbation hypotheses10-37
10.4.2.2 Base-flow cooling, diffusion and chemical
source term . . . 10-41
10.4.2.3 Equilibrium-perturbation hypothesis . . . 10-44
10.4.2.4 Supersonic modes . . . 10-47
10.4.2.5 Transport model in CPG . . . 10-50
10.4.3 Free-stream boundary condition . . . 10-51
10.4.4 Conclusions . . . 10-51
References . . . 10-55
11 Decoupling ablation-transition problems 11-1
11.1 Flow assumptions . . . 11-2
11.2 Boundary conditions . . . 11-3
11.3 Results . . . 11-5
11.3.1 Wall profiles from the CNE11-Abl case . . . 11-7
11.3.3 Vibrational excitation and air chemistry effects . . . 11-14
11.3.4 Surface chemistry effects . . . 11-18
11.3.5 CPG assumption with different transport models . . . 11-21
11.3.6 Influence of the transport models on the fully-ablating case 11-22
11.3.7 N-factor budgets of the physical phenomena . . . 11-23
11.3.8 Linearized Rankine-Hugoniot relations . . . 11-26
11.4 Conclusions . . . 11-26
References . . . 11-30
12 Conclusions 12-1
References . . . 12-9
IV Appendices
A General relations between species and mixture quantities A-1
References . . . A-4
B Equilibrium system of equations B-1
B.1 Equilibrium-composition derivatives with respect to temperature,
pressure or elemental fractions . . . B-2
B.2 The air-5 equilibrium system . . . B-6
B.3 The air-11 equilibrium system . . . B-7
B.4 Verification . . . B-8
References . . . B-8
C Coefficients for the transport properties C-1
C.1 Coefficients for the polynomial approximation of
Chapman-Enskog’s theory for transport properties . . . C-1
C.2 Coefficients for Brokaw and Yos’ simplification of Chapman &
Enskog’s theory for transport properties . . . C-3
References . . . C-4
D Requirements for the existence of self-similar solutions to the
boundary-layer equations D-1
D.1 CPG with a viscosity power law and with qe = qe(ξ) . . . D-2
D.2 CPG with other viscosity laws and qe = qe(ξ) . . . D-5
D.3 TPG with qe = qe(ξ) . . . D-6
D.4 LTE and LTEED with qe = qe(ξ) . . . D-6
D.5 TCFG with qe = qe(ξ) . . . D-7
D.6 Surface thermal boundary condition . . . D-7
D.7 Surface mass injection . . . D-8
D.8 y-vector rebuilding for βH �= 0 . . . D-8
D.9 Summary . . . D-9
E Rebuilding of the wall-normal velocity from a boundary-layer solutionE-1
F Base-flow wall-normal velocity treatment in LST when using the
AMSBC F-1
References . . . F-2
G Integration matrices of arbitrary-order accuracy G-1
References . . . G-2
H Neutral curve Newton-Raphson solver H-1
References . . . H-3
I Non-dimensional form of the boundary conditions I-1
J Gas mixtures and reaction rate constants J-1
J.1 Air-2 . . . J-1 J.2 Air-2-He . . . J-1 J.3 Air-2-Ne . . . J-1 J.4 Air-2-Ar . . . J-1 J.5 Air-2-CO2 . . . J-2 J.6 O-2-Bortner . . . J-2 J.7 Air-5-Park85 . . . J-2 J.8 Air-5-Park90 . . . J-3 J.9 Air-5-Park01 . . . J-4 J.10 Air-5-Stuckert . . . J-5 J.11 Air-5-Bortner . . . J-6 J.12 Air-11-Park93 . . . J-8 J.13 AirC-6-Mortensen . . . J-11 J.14 AirC-11-Mortensen . . . J-12 References . . . J-16 K Species-parameters tables K-1 K.1 Thermal parameters . . . K-1 K.2 Transport parameters . . . K-12 K.3 Collision parameters . . . K-14
K.3.1 Stuckert’s [16] curve fits (Eq. 4.56) . . . K-14
K.3.2 Gupta et al.’s [15] curve fits (Eq. 4.55) . . . K-16
K.3.3 Novel curve fits (Eq.4.57) to recent data . . . K-18
K.3.3.1 Air neutral species (N, O, NO, N2, O2) . . . K-18
K.3.3.2 Air neutral (N, O, NO, N2, O2) and charged
species (N+, O+, NO+, N+
2, O+2) . . . K-20
K.3.3.3 Air neutral species (N, O, NO, N2, O2) and
ar-gon (Ar) . . . K-26
K.3.3.4 Air neutral species (N, O, NO, N2, O2) and
car-bon species (CO, CO2, CN, C, C2, C3) . . . K-27
L Collision-integral fits L-1
L.1 Air collisions of neutral-to-neutral species (N, O, NO, N2, O2) . . L-1
L.2 Air collisions of charged-to-charged species (N+, O+, NO+, N+ 2,
O+
2, e-) . . . L-4
L.3 Air collisions of ion-to-neutral species (N+, O+, NO+, N+
2, O+2 to
N, O, NO, N2, O2) . . . L-7
L.4 Air collisions of electron-to-neutral species (e- to N, O, NO, N
2, O2)L-12
L.5 Air collisions of argon-to-neutral species (Ar to N, O, NO, N2, O2) L-14
L.6 Collisions of neutral air and carbon species (N, O, NO, N2, O2 and
