Aerospace Science and Technology 43 (2015) 360–371

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Aerospace Science and Technology

www.elsevier.com/locate/aescte

Numerical analysis of hypersonic flows around blunt-nosed models

and a space vehicle

Heriberto Saldivar Massimi a,1, Hua Shen a,2, Chih-Yung Wen a,∗,3, Yen-Sen Chen b,4, Shen-Min Liang c,5

a Department of Mechanical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kongb National Space Organization, Hsinchu Science Park, Hsinchu 30078, Taiwanc Department of Computer Application Engineering, Far East University, Tainan 744, Taiwan

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 September 2014Received in revised form 18 March 2015Accepted 20 March 2015Available online 25 March 2015

This work addresses the problem of the aerothermodynamics of hypersonic nonequilibrium flows over blunt nosed models and space vehicles with rarefaction effects. First, the in-house Navier–Stokes solver, UNIC-UNS code, with the slip boundary condition and finite-rate chemistry is used to simulate the hypersonic flows over a blunt nosed model and the simplified European eXPErimental Re-entry Test-bed (EXPERT) model V4.4. Next, hypersonic flows over the whole EXPERT 3D model, which correspond to the expected descent trajectory with allowance for rarefaction and thermochemical nonequilibrium are simulated. By comparing with the Direct Simulation Monte Carlo (DSMC) method, it is observed that the UNIC-UNS code is reliable in simulating hypersonic flows with rarefaction and thermochemical non-equilibrium effects. A detailed analysis of the aerothermodynamics for EXPERT for a wide range of flow regimes is also provided by utilizing the numerical flow visualization. The present numerical simulations provide some important data for EXPERT, which cannot be easily derived by experiments. This study aims to work as a precursor for future studies and to provide to the scientific community with quality data that can be used to improve tools for the design of hypersonic vehicles.

© 2015 Elsevier Masson SAS. All rights reserved.

1. Introduction

Following renewed interest in manned missions to Mars and the moon as well as increasing success in the development of new thermal protection materials offering higher possibilities of developing space vehicles capable of withstanding the higher tem-peratures caused by higher speed reentries, the problem of study-ing specific features of high-velocity flows has brought renewed attention to the field of hypersonic aerothermodynamics. Several space agencies, e.g., NASA and ESA (Crew Exploration Vehicle [1],

* Corresponding author.E-mail address: cywen@polyu.edu.hk (C.-Y. Wen).

1 Graduate Student, Department of Mechanical Engineering, The Hong Kong Poly-technic University, 11 Yuk Choi Rd, Hung Hom, Hong Kong.

2 Postdoctoral Research Associate, Department of Mechanical Engineering, The Hong Kong Polytechnic University, 11 Yuk Choi Rd, Hung Hom, Hong Kong.

3 Professor, Department of Mechanical Engineering, The Hong Kong Polytechnic University, 11 Yuk Choi Rd, Hung Hom, Hong Kong.

4 Senior Research Fellow and Suborbital Rocket Experiment Program Director, Na-tional Space Organization, 8F, 9, Prosperity 1st Rd., Hsinchu Science Park, Hsinchu, Taiwan 30078.

5 Professor. Department of Computer Application Engineering, Far East University, No. 49, Chung Hua Rd., Hsin-Shih, Tainan County 744, Taiwan.

LAPCAT I and LAPCAT II [2]) as well as private companies (ZEHST by EADS [3] and Dragon by Space Exploration Technologies Corpo-ration [4]) are currently designing several hypersonic vehicles that, hopefully, will become the next generation of high-speed vehicles.

The development of future generation of space vehicles requires a complete and detailed knowledge of their aerothermodynam-ics along with the complete descent trajectory. For this reason, it is necessary to study phenomena associated with rarefaction and thermochemical non-equilibrium of the gas in a hypersonic flow. Such a study could be of help in accurately predicting the in-flight heat flux, pressure and shear stresses from which the thermal load, aerodynamic forces and moments can be calculated. The geometry of the vehicle, and in particular, of the nose and the leading edges of wings and other aerodynamic surfaces, is of a critical consider-ation in vehicle design.

However, for nearly every ground-based facility, typical param-eters (Mach number, Ma; and Knudsen number, Kn) associated with flow around the space vehicle at high altitudes are lim-ited to certain ranges. Therefore, computational fluid mechanics (CFD) has become a necessary supplementary tool in studying these phenomena [5]. Further, the flight-testing and reproduction of these varied flow conditions in ground-based laboratories are

http://dx.doi.org/10.1016/j.ast.2015.03.0171270-9638/© 2015 Elsevier Masson SAS. All rights reserved.

H. Saldivar Massimi et al. / Aerospace Science and Technology 43 (2015) 360–371 361

Nomenclature

a speed of soundC heat capacityCf pressure coefficientCh force coefficient in the x directionCp pressure coefficientD nose diameterH enthalpy or energyK thermal conductivityk,ε turbulence parametersmw mixture molecular weightp surface pressurep∞ freestream pressureq convection heat fluxQ volumetric heat sourceQ t heat sourceℜ universal gas constantTs Solid Temperature

v stoichiometric coefficientsωn species production source termX axial locationαn nth species mass fractionγ ratio of specific heat% dimensionless stand-off distanceϑr turbulence kinetic energy production rateµl laminar viscosityµt turbulence eddy viscosityµe effective viscosityρb body densityρb equilibrium densityρs density right after the normal shockwaveσ Prandtl number/Schmidt numberτw wall shear stressΦ energy dissipation functionΩ dimensionless chemical reaction parameter

Fig. 1. Geometry of the EXPERT (V4.4) capsule.

both expensive and technically challenging. Hence, computational models play extremely important roles in the development of hy-personic vehicles.

Nevertheless, the lessons learned from past flight-test pro-grams, such as the ESA Atmospheric Re-entry Demonstrator (ARD) have underlined the need for more accurate and extensive hyper-sonic flight data. This is particularly true with the characterization of hypersonic phenomena such as high temperature and chem-istry effects, gas-surface interaction, catalysis and oxidation. The European Experimental Reentry Test bed (EXPERT) is a flight ve-hicle aiming to collect the aerothermodynamic (ATD) flight data needed in validating design tools, ground test facilities and ver-ification techniques [5]. Therefore, EXPERT has been developed to benefit future atmospheric re-entry activities ranging from cargo to human orbital transportation systems as well as reusable launch-ers and scientific probes. The EXPERT configuration (see Fig. 1) is composed of a conical body with a cone angle of 12.5 , truncated by planes at an angle of 8.35 to the axis of symmetry featuring four control flaps deflected by 20; and a nose with a local radius of 0.6 m; the nose–cone junction is described by a clothoid.

Due to the expected high temperatures around the capsule sur-face, EXPERT was designed to have two main structures; a hot

Fig. 2. Evolution of the velocity and enthalpy of the EXPERT capsule along its de-scent trajectory [5].

outer structure and a cold structure, which are decoupled from each other. The three main parts of the hot body structure are: the nose cap and flaps made of ceramic composite materials, and the general body made of oxide dispersion strengthened super alloy.

It was planned that EXPERT will be launched with a Russian converted ICBM Volna missile from a submarine, flying a suborbital ballistic trajectory from the Pacific Ocean to a landing site located on the peninsula of Kamchatka. It will be injected with an initial flight path angle of −5.5 and a velocity of 5 km/s at an altitude of 100 km. The EXPERT test window will last around 140 s until the drogue parachute is deployed, allowing a landing speed lower than 10 m/s [5]. The evolution of the velocity and enthalpy are shown in Fig. 2. The mission was designed such that the flight speeds are compatible with conditions that can be achieved from ground facilities, thus allowing extensive comparisons between flight and ground data, which is of primary importance to validate theoret-ical models. Notably, as of 2012, Russia has withdrawn from the EXPERT project and the launch by its Volna rocket was canceled. The launch is now expected to be carried out by the ESAs Vega small-satellite launcher [6].

Several studies have been performed on the EXPERT capsule. For example: Vashchenkov et al. [7] ran a series of numerical sim-ulations for different angles of attack and rolling of the capsule using nitrogen at altitude of 107 km down to 85 km; Schettino, et al. [8] performed an analysis for aerodynamic characteristics at an

362 H. Saldivar Massimi et al. / Aerospace Science and Technology 43 (2015) 360–371

angle of attack of 3 at a different range of Mach numbers using the structured code H3NS and by TAS-I using NAXTDI unstructured code, they paid special attention to the effect of wall catalycity and the effects of transition imposed at the nose–cone junction; Mas-sobrio et al. [9] ran simulations the for EXPERT versions 4.2 and 4.3 at an angle of attack of 5.2 and a speed of 6000 m/s.

Although the Direct Simulation Monte Carlo (DSMC) method and several continuum CFD methods have both been used to model the EXPERT capsule along its descent trajectory a study of the capsule’s final version in the continuum and transitional regime to the best of the current authors’ knowledge have not been studied.

This work extends the study previously done by the authors [10] and presents the results obtained by the in-house Navier–Stokes (NS) solver, UNIC-UNS [11], which is based on a high-order finite difference method, with the slip boundary condition and finite-rate chemistry. A pressure-based predictor plus a multi-corrector solution method is employed in the code to make it suitable for all speed flows. First, the hypersonic nonequilibrium flow over a simple blunt-nosed model is studied in the interest of reliable validation. Next the code is employed to simulate the hypersonic flows around the EXPERT capsule, which correspond to the expected descent trajectory with allowance for rarefaction and thermochemical nonequilibrium. Section 2 provides a brief description of the numerical method, comprising an explanation of the UNIC-UNS code, the turbulence model, simulation meth-ods, slip and temperature boundary conditions and the finite-rate chemistry model. Section 3 covers the simulation models and test conditions used in the current research. Sections 4 and 5 discuss the results obtained. In Section 4, simulations of hypersonic flows about a two-dimensional (2D) cylindrically blunted flat plate are carried out so the slip boundary condition is validated and the re-sults are compared with those of DSMC method from Bondar et al. [12]. For validation of the chemical non-equilibrium effect us-ing the UNIC-UNS code, simulations are run using an axisymmetric hemi-spherical-nosed cylinder and finite-rate chemistry and com-pared with simulations performed by Tchuen and Zeitoun [13]. Section 5 shows the results of hypersonic flows around the EX-PERT Capsule along its descent trajectory, a set of axisymmetric cases is used to calculate the different wall temperatures of the EXPERT capsule along its descent trajectory. As a further validation study, a comparison of the dimensionless stand-off distance of the EXPERT capsule was performed based on the correlation of stand-off distance reported by Wen and Hornung [14] and Belouaggadia, Olivier and Brun [15], then three dimensional simulations of the EXPERT capsule with no roll angle and an angle of attack (AoA) of −5.5 are shown. The surface parameter along the lower center-line of the capsule and the chemical reactions around the capsule are discussed. Finally, Section 6 offers the major conclusions and some general remarks.

2. Numerical methods

2.1. UNIC-UNS code

The in-house UNIC-UNS code [11], a multi-grid hybrid unstruc-tured solver for all-speed flows is used for NS computations of thermally and chemically non-equilibrium airflow. The code has been used successfully for the analysis of a solid-core nuclear ther-mal engine thrust chamber [16], reacting flow computations [17], prediction of low frequency combustion instability in a model ram-jet combustor [18], and reacting and non-reacting flow simulation of film cooling in 2-D supersonic flows [19].

The basic governing equations of the code are the following multicomponent transport expressions, which can be written, in

the general form

∂ρU∂t

+ ∂

∂xi

(ρuiU + µe

∂U∂xi

)= S(U) (1)

where ρ and U = [1, u, v, w, H, k, ε, αn]T stand for the fluid den-sity and the vector of flow primitive variables consisting of con-tinuity, momentum, energy, turbulence model and species mass-fraction equations respectively. This general form of transport equations has one exception that fluid temperature instead of en-thalpy is used for the diffusion (heat conduction) terms in the energy equation. The source terms, S(U), for the momentum, en-ergy, turbulence model and species mass fraction equations in 3D space x can be written as

S(U) =

⎡

⎢⎢⎢⎢⎢⎢⎣

0− ∂ p

∂x j+ ∂

∂xi(µe

∂ui∂x j

) − 23

∂∂x j

(µe∂ui∂xi

)DpDt + Φ + Q tρ(ϑr − ε)

ρ εk [C1ϑr − C1ε]

ωn, n = 1, N

⎤

⎥⎥⎥⎥⎥⎥⎦(2)

where µe = (µl + µt)/σ is the effective viscosity estimated as the sum of the laminar viscosity and the turbulence eddy viscosity values divided by the turbulence Prandtl number (for energy equa-tion) or Schmidt number (for the other equations), σ .Φ , Q t and ωn are the energy dissipation function, heat source and the species source term respectively. ϑr stands for the turbulence kinetic en-ergy production rate, which can be written as

ϑr = µt

ρ

12

(∂u j

∂xi+ ∂ui

∂x j

)2

− 23

(∂uk

∂xk

)2(3)

The turbulence modeling constant C1 takes the functional form C1 = 1.15 + 0.25 ϑr

ε and C2 is a modeling constant equaling 1.90.Turbulence Schmidt numbers for the k and ε equation, are

0.8927 and 1.15 respectively. The turbulence Schmidt number for the species mass fraction equation is assumed to be 0.9. The same value is assumed for the energy equation of the turbulent Prandtl number (0.7 is used for laminar flows).

To account for compressibility effects on the turbulent model, two methods of model correction are employed in the code. They are: (1) the k-equation correction, in which the dissipation term-in the k-equation is replaced by the following term

ε(1 + Ma2

t)

(4)

where,

Mat =√

ksound speed

and (2) an ε-equation correction, in which C1 in the ε-equation is modified by the following correction factor

(1 + 0.08Ma0.25) (5)

where Ma is the local Mach number.The above governing equations are closed by an equation of

state:

ρ = pℜT /mw

(6)

where, ℜ, T and mw stand for the universal gas constant, fluid temperature and the mixture molecular weight, respectively.

H. Saldivar Massimi et al. / Aerospace Science and Technology 43 (2015) 360–371 363

2.2. Simulation methods

The basic numerical algorithm of the UNIC-UNS code uses a finite difference method for solving the nonlinear transport equa-tions expressed by Eq. (1). A central plus adaptive second-order and fourth-order dissipation scheme is used to approximate the convection terms. Viscous fluxes and source terms are discretized by a second-order central difference. A pressure-based predictor plus a multi-corrector solution method is employed in the code so that it can be applied to all speed flows. Such a temporal dis-cretization allows the discrete finite difference equations to be ar-ranged into a delta form for time-marching integration. An implicit centered time-marching scheme is used for performing time accu-rate computations. More details can be found in [20].

Moreover, the code is parallelized with Message Passing Inter-face (MPI). In performing parallel computation, the domain de-composition method is currently used in the code. Both the re-cursive coordinate bisection (RCB) and recursive graph bisection (RGB) methods available in a publicly available package developed at University of Minnesota, METIS [21], are employed for efficient decomposition. The ALPS cluster, a system that estimates the ag-gregate performance of over 177 TFLOPS, uses the AMDR Opteron 6100 processors, and has a total of 8 computer clusters, 1 large memory cluster, and over 25,600 computer cores for numerical simulations was used. Total CPU hours equal to about 10,240 were used in each simulation of hypersonic flow over the complete 3D EXPERT V4.4 model. The cluster was provided by the National Cen-ter for High-performance computing in Hsinchu, Taiwan R.O.C.

2.3. Slip and temperature boundary condition

The boundary conditions [22] for the wall slip velocity, uw , and the wall temperature jump, Tgas − T w , implemented in the in-house code are as follows:

uw ≈(

2f

− 1)

l(

dudy

)

w(7)

Tgas − T w =(

2α

− 1)

2γ

(γ + 1)Prl(

dTdyw

)(8)

where l = 32 ( τw

ρa ), f = 1, τw is the wall shear stress, a is the speed of sound, Tgas and T w are the translational temperature of the gas and the wall temperature, respectively, α = 1, γ the ratio of spe-cific heat, and Pr the Prandtl number.

2.4. Finite-rate chemistry model

A general system of chemical reactions can be written in terms of its stoichiometric coefficients (vij and v ′

i j) and the associated i-th chemical species (mi ) in the j-th reaction as,∑

i

vi jmi !∑

i

v ′i jm

′i (9)

The net rate of change in the molar concentration of species i due to reaction j, Xij , can be written as:

xij =(

v ′i j − vij

)[K f j

∏(ραi

mwi

)vi j

− Kbj

∏(ραi

mwi

)v ′i j]

(10)

and the species production rate (in terms of mass fraction), wi , is calculated by summing over reactions. That is, wi = mwi

∑j Xi j

where mwi is the molecular weight of species i, αi the mass frac-tion of species i, K f j is the forward rate of reaction j given in the Arrhenius form as K f j = A j T B j exp(− E j

RT ); Kbj = K f jKej

is the back-

ward rate of reaction j, where Kej = ( 1RT )

∑(v ′

i j−vij)EXP[∑( f ′i v ′

i j −

f i vi j)] is the equilibrium constant. Constants A, B and E are given by experimental correlation and f i is the Gibbs free energy of species i.

A penalty function method or a point-implicit integration pro-cedure can be employed to ensure the basic element conservation constraints at the end of each time marching step. This is a crucial requirement for the numerical stability and accuracy of the CFD solution.

For airflows involving ionization, such as those considered here, primary chemical species and the chemical reactions considered for the flow are as follows:

O2 + M ! O + O + M

N2 + M ! N + N + M

NO + M ! N + O + M

NO + O ! N + O2

O + N2 ! N + NO

O + O+2 ! O2 + O+

N2 + N+ ! N + N+2

O + NO+ ! NO + O+

N2 + O+ ! O + N+2

O2 + NO+ ! NO + O+2

NO+ + N ! N+2 + O

O + N ! NO+ + e−

O + O ! O+2 + e−

N + N ! N+2 + e−

O + e− ! O+ + 2e−

N + e− ! N+ + 2e−

where M represents any species that acts as a collision partner in the reaction.

3. Simulation models and test conditions

3.1. Hypersonic flows over a 2D cylindrically blunted flat plate and an axisymmetric hemi-spherical-nosed cylinder

With the view to validating the UNIC-UNS code at different Knudsen numbers, flows over a 2D cylindrically blunted flat plate were simulated for cases one to four in Table 1. The velocity and temperature profiles, the skin friction coefficient, Stanton number and pressure distributions were compared with those of DSMC by Bondar et al. [12]. Argon was adopted as the test gas. A schematic diagram of the model is shown in Fig. 3. Radius R was set as the reference length equal to 1 unit and the plate length of 10R . The computational domain was bounded between the supersonic in-let and the outlet. A 2D structured half-model was used in all test cases. The values of R were adjusted to meet the required Knudsen number (see Table 1).

To validate the UNIC-UNSs capability in performing the chem-ical non-equilibrium flow simulations in a continuum regime, a Mach 18 dry air flow (Case 5 of Table 1) over an axisymmet-ric hemi-spherical-nosed cylinder was simulated. The simulation model (see Fig. 3) and the freestream conditions are taken from the numerical works done by Tchuen and Zeitoun [13], and the ex-perimental observations reported by Rose and Stankevics [23]. Dry Air and Park’s two-temperature chemical model [24] were used.

364 H. Saldivar Massimi et al. / Aerospace Science and Technology 43 (2015) 360–371

Table 1Test matrix for the validation of UNIC-UNS.

Case Test gas Kn R (m) Type Ma∞ P∞ (Pa) ρ∞ (kg/m3) T∞ (K)

1 Argon 0.1 0.010 2D 5 5.29 1.142E−4 219.5852 Argon 0.50 0.0023 2D 5 5.29 1.142E−4 219.5853 Argon 0.1 0.010 2D 10 5.29 1.142E−4 219.5854 Argon 0.50 0.0023 2D 10 5.29 1.142E−4 219.5855 Dry air 0.00205 0.0066 3D 18 432.2 5.973E−5 252.000

Fig. 3. Computational domain and boundary conditions applied to the spherical-nosed cylinder and the cylindrically blunted flat plate.

Fig. 4. Reynolds numbers and Knudsen numbers of the EXPERT capsule vs. altitude.

3.2. Hypersonic flow around the EXPERT capsule

The capsule’s Knudsen and Reynolds numbers vary according to the altitudes (see Fig. 4). To calculate the Reynolds numbers (Re) and the Knudsen numbers (Kn), the nose diameter, D = 1.2 m, and the nose radius, D/2 were adopted as the reference lengths, re-spectively. As shown in Fig. 4, during the descent trajectory, the freestream Knudsen number varies from free-molecular flow to continuum flow and the Reynolds number varies from laminar flow to turbulent flow.

To better predict the heat transfer rate around the EXPERT V4.4 capsule (Fig. 1), wall temperatures need to be estimated. Consider-ing optimum use of computational resources, the 1-D conjugated

Fig. 5. Computational domain and boundary conditions applied in the axisymmetric simulations of the EXPERT V4.4’s cross-section for the prediction of the capsule’s wall temperature. AOA = 0 .

Table 2Material properties for the nose and flaps region.

Thermal conductivity (W/m/K)

Density (kg/m3)

Specific heat (J/g/K)

125 2700.0 700

Table 3Material properties for the hot surface region.

Temperature (K)

Thermal conductivity (W/m/K)

Density (kg/m3)

Specific heat (J/g/K)

373 12 8240.0 440.0473 23 8240.0 500.0773 30 8240.0 620.0

1023 37 8240.0 720.01273 42 8240.0 820.01473 42 8240.0 890.0

heat transfer model can be written as

∂ρC Ts

∂t= ∂

∂x j

(K

∂Ts

∂xi

)(11)

where C , K and Ts stand for heat capacity, thermal conductiv-ity and solid temperature, respectively. The wall heat conduction equation was solved through an iterative procedure until con-vergence was achieved for each time step. The heat conduction model for the wall points was applied to a simplified EXPERT V4.4 model to estimate wall temperatures. It was assumed a wall thickness of 1.5 cm. The initial conditions for the temperatures on the wall and inside the capsule were set up to be the same as the freestream temperature and 300 K, respectively. Axisymmet-ric simulations were performed on the computational domain of the EXPERT capsule’s cross-section (see Fig. 5). The computational domain was composed of 44,800 structured meshes bounded be-tween the supersonic inlet and the outlet. Tables 2 and 3 show the thermal conductivity, density and specific heat values adopted

H. Saldivar Massimi et al. / Aerospace Science and Technology 43 (2015) 360–371 365

Table 4Test matrix for EXPERT capsule.

Altitude (km) Ma∞ Freestream speed (m/s)

P∞ (Pa) ρ∞ (kg/m3) T∞ (K)

20 6.66 2000 5502 8.90E−2 217.0040 15.62 5000 289.8 4.00E−3 250.0060 15.86 5000 21.89 3.09E−4 247.0270 16.83 5000 5.2 8.28E−5 219.5880 17.69 5000 1.1 1.85E−5 198.6090 18.3 5000 0.18 3.38E−6 187.00

Fig. 6. The pressure coefficient distribution along the lower centerline at an altitude of 90 km simulated by different mesh sizes.

for each section of the capsule wall (nose, flaps and hot body sur-face) [25].

Once the wall temperature was calculated, a complete 3D sim-ulation of the EXPERT Capsule V4.4 with an angle of attack of −5.5 was performed for the prediction of the capsule’s aero-thermodynamic performance. Table 4 shows the test matrix cor-responding to the expected descent trajectory of the EXPERT V4.4 capsule from an altitude of 90 km down to 20 km with allowance for rarefaction and thermochemical nonequilibrium [5]. The test gas was dry air with a freestream composition of nitrogen and oxygen mass fractions, 0.778 and 0.222, respectively. The chemi-cal kinetic model used was that of Park’s two-temperature model [24]. For the cases of 90 km down to 40 km, the model was posi-tioned with an angle of attack of −5.5 and a freestream velocity of 5000 m/s whereas, for the case of 20 km, the angle of attack was set at −5.5 and freestream velocity of 2000 m/s. The com-putational domain applied was bounded between the supersonic inlet and the outlet. A hybrid mesh of 9,326,336 cells composed the computational domain.

The grid-independence tests were carefully examined for flows over a 2D cylindrically blunted flat plate, an axisymmetric hemi-spherical-nosed cylinder and the EXPERT Capsule. This was achieved by running a series of simulations by increasing the mesh size by a factor of 1.5 until the difference in the surface properties; pressure coefficient, Cp, skin friction coefficient, Cf , and Stanton number, Ch, between mesh models was smaller than 2%. Fig. 6 shows the pressure coefficient distribution along the lower centerline at an altitude of 90 km simulated by different mesh sizes. Skin fric-tion coefficient and Stanton number distributions also show similar convergence with the mesh size as the pressure coefficient distri-bution.

4. Results associated with hypersonic flows over a 2D cylindrically blunted flat plate and an axisymmetric hemi-spherical-nosed cylinder

4.1. Validation of slip boundary conditions

Rarefied Argon flows at Mach numbers Ma = 5 and Ma = 10around a 2D cylindrically blunted thick flat plate (Cases 1–4) were calculated by the UNIC-UNS code and compared with the DSMC method. The temperature fields for Ma = 5 Argon flows are shown in Fig. 7 with Kn = 0.1 and 0.5. Similar results with Kn = 0.1 and 0.5 were obtained for Ma = 10 flows but not shown here. Fig. 8, shows in great detail the temperature profile along the stagnation streamline for different Mach and Knudsen numbers. As shown in Figs. 7 and 8, although the shock-wave thicknesses obtained by the UNIC-UNS code and the DSMC method are appreciably different, especially at higher Knudsen numbers, the temperature profiles of the UNIC-UNS code approached that of the DSMC method when the fluid approached the wall.

As previous studies have shown [26–28], for hypersonic flows with higher Knudsen numbers; the first continuum assumption breakdown was observed within the shock wave itself. It’s known that the traditional NS-based CFD cannot accurately predict hy-personic shock structures [29,30]. However, the addition of slip velocity and temperature jump boundary conditions greatly im-proved agreement between the NS-based CFD and DSMC on the rest of the flow field and the resulting surface aerothermodynam-ics properties. Typical comparisons of the pressure coefficient (Cp), skin friction coefficient (Cf ) and Stanton numbers (Ch), between the NS simulations of case 2 with the slip boundary conditions, and the published DSMC results [7] are shown in Fig. 9. It can be seen that, despite relatively small discrepancies found in the skin friction and heat transfer comparisons around the expanding shoulder region (where the half-cylinder is connected to the flat plate), the overall agreement between NS and DSMC simulations are good in regards to the pressure, skin friction and heat trans-fer distributions, indicating that the slip velocity and temperature jump boundary conditions are all well justified. The definitions of the pressure coefficient, skin friction coefficient and Stanton num-ber are as follows

Cp ≡ p − p∞12ρ∞V 2∞

(12)

Cf ≡ τw12ρ∞V 2∞

(13)

Ch ≡ q12ρ∞V 3∞

(14)

where p is the wall surface pressure, p∞ is the freestream pres-sure, ρ∞ is the freestream fluid density and q is the convection heat flux.

As the Knudsen number decreases and flow approaches contin-uum, the results calculated by the two approaches become closer. The NS equations with the slip boundary conditions are applicable at Kn = 0.5 and smaller. As the Knudsen number decreases below

366 H. Saldivar Massimi et al. / Aerospace Science and Technology 43 (2015) 360–371

Fig. 7. Comparison of temperature fields between UNIC-UNS and DSMC results for cases 1 and 2 in Table 1. Ma = 5, Kn = 0.1 (left) and Kn = 0.5 (right).

Fig. 8. Comparison of temperature profiles along the stagnation streamline between UNIC-UNS and DSMC results, for cases 1 to 4 in Table 1. Left: Ma = 5 and right: Ma = 10.

Fig. 9. Comparison of surface aerothermodynamic characteristics between UNIC-UNS and DSMC results, for Case 2 in Table 1, Ma = 5 and Kn = 0.5: Pressure and friction coefficients (left) and Stanton number (right).

H. Saldivar Massimi et al. / Aerospace Science and Technology 43 (2015) 360–371 367

Fig. 10. Comparison of temperature profiles along the stagnation streamline be-tween UNIC-UNS results and those from Tchuen and Zeltoun [13]. (Case 5 in Table 1at Mach 18) where X/D = 0 corresponds to the stagnation point.

0.1, the results predicted by the UNIC-UNS code are almost identi-cal to the solution from the Boltzmann equation.

4.2. Validation for chemical non-equilibrium

Case 5, with the flow conditions based on the simulations done by Tchuen and Zeitoun [13], and the experimental obser-vations reported by Rose and Stankevics [23], was run by the UNIC-UNS code to validate its capability to predict hypersonic chemical dissociation of flow fields. Fig. 10 shows the tempera-ture profile along the stagnation streamline of an axisymmetric hemi-spherical-nosed cylinder. It can be seen that directly be-hind the shock, the temperature increases abruptly and reaches its maximum value. It can be observed that, as the velocity de-creases when the flow approaches the stagnation point, the gas molecules have more time to dissociate, therefore bringing the temperature down to even lower values. Though the maximum temperature value shows a slight difference of 2.65% when com-pared with the results of Tchuen and Zeitoun [13], the temperature profiles behind the bow shock waves in both cases are similar. This good agreement proves the validity of the NS equations with the Parks two-temperature chemical model and the velocity slip and temperature jump conditions in simulating the hypersonic chem-ical dissociating flows. Note that the simulations of Tchuen and Zeitoun [13] have been well validated with the experiments of Rose and Stankevics [23]. A comparison of the stand-off distances between the UNIC-UNS simulations and the experimental results of Rose and Stankevics [23] show a difference of 1.259%. Good agreement is observed again and the in-house UNIC-UNS code is capable of providing accurate predictions of the shock positions for non-equilibrium hypersonic flows over spherical-nosed cylin-ders.

5. Results associated with hypersonic flows around the EXPERT V4.4 capsule

5.1. Stand-off distance validation

Correlation results associated with the stand-off distance of a hypersonic chemical dissociating gas flow over a sphere have been described by Wen and Hornung [14] and Belouaggadia, Olivier and Brun [15] and widely applied in related studies. Wen and Hor-nung [14] described that the dimensionless stand-off distance of non-equilibrium dissociating flow over spheres, in the form of

Fig. 11. Temperature distributions along the centerline over the EXPERT capsule at an altitude of 70 km in the simplified axisymmetric model.

% ≡ %D

ρsρ∞ , could be correlated well with a dimensionless chem-

ical reaction parameter Ω ≡ Dρsu∞ ( dρ

dt )s , where the subscript, s, indicates the conditions right after the normal shock wave. The physical significance of Ω has been clearly explained by Wen and Hornung [14] as

Ω = Energy absorption rate by chemistryInput rate of freestream kinetic energy

And % is related to the average density. Therefore, the exact details of the density profile are not important.

When Ω is small, the flow is slightly dissociated and the den-sity on the body ρb is smaller than the equilibrium density ρe . % was derived by Wen and Hornung [14] as follows

% = 1

Ω

[−1 + (1 + 2LΩ)1/2] (15)

where L is 0.41 for spheres. % depends only on Ω . However, when the density on the body equals the equilibrium density, Eq. (15)no longer holds, because the equilibrium value of the density now enters into the calculation of average density. Forming the aver-age density from the linear profile up to the point where density reaches ρe and a constant density part with ρ = ρe thereafter, the dimensionless stand-off distance was derived as

% = ρs

ρe

[L + 1

2Ω

(ρs

ρe− 1

)2](16)

As a validation study, the currently simulated dimensionless stand-off distance of the EXPERT V4.4 capsule was compared with the estimate from Wen and Hornung’s theory [14] (Eqs. (15)–(16)). The nose of the capsule was considered to be a sphere of radius, 0.6. The flow conditions are listed in Table 4. From the results (see Ta-ble 5), it is seen that the in-house UNIC-UNS code is capable of providing good predictions of shock stand-off distances associated with hypersonic dissociating flows over the sphere-nosed EXPERT model.

5.2. Prediction of flow features and aerothermodynamic characteristics of the EXPERT capsule

The simplified axisymmetric EXPERT V4.4 model described in Section 3.2 was used to estimate the wall temperatures at dif-

368 H. Saldivar Massimi et al. / Aerospace Science and Technology 43 (2015) 360–371

Table 5Comparison of normalized stand-off distances (%/D) for the EXPERT capsule at different altitudes with Wen and Hornung’s theory [14]. AOA = 0.0 .

Altitude Ω %/D Difference %Theory Calculated (Eqs. (15)–(16))

20 km 22.2 0.0643 0.0654 1.6%40 km 81.74 0.0349 0.0363 4.1%60 km 82.07 0.0304 0.0316 4.0%70 km 83.15 0.0282 0.0292 3.4%80 km 91.4 0.0275 0.0289 5.1%90 km 92.0 0.0241 0.0260 2.6%

Fig. 12. Temperature a) and pressure b) contour for the EXPERT capsule at an altitude of 40 km.

Table 6Highest temperatures in the nose, flap and hot body surface sections of the EXPERT (V4.4) capsule in the simplified axisymmetric model.

Altitude (km)

Nose temperature (K)

Hot body temperature(K)

Flaps temperature (K)

20 608.44 1639.47 577.9940 424.927 603.127 378.1460 331.28 363.27 286.7770 316.36 344.18 273.1080 268.03 339.82 236.7590 240.28 286.46 232.11

ferent parts on the EXPERT capsule along its descent trajectory. Fig. 11 shows a typical temperature distributions in different sec-tions along the centerline of the EXPERT capsule. The surface tem-perature at the nose tip is lower than those along the conical sur-face. This behavior is a consequence of the different solid proper-ties as well as the conjugate heating model. Notably, the nose cap of EXPERT is a ceramic matrix composited piece, which has a much higher thermal conductivity (K = 125 W/m/K from Table 2) than the body wall made of a metallic super alloy (K = 12 W/m/K–42 W/m/K from Table 3). The heat flux of the solid is calculated by qs = −K ∂Ts

∂x . Under the condition of steady state, heat flux into the solid is equal to that from the gas. It was assumed a con-stant wall thickness of 1.5 cm and a constant temperature of 300 K on the inner wall as the boundary condition. When qs is fixed, higher thermal conductivity will lead to a lower temperature on the surface. Table 6 shows the highest temperatures in the nose, flap and hot body surface sections of the EXPERT (V4.4) capsule in the simplified axisymmetric model shown in Fig. 5 at differ-ent altitudes. The higher temperature observed at the low altitude than at the high altitudes (although the flight Mach numbers are lower at low altitudes), is due to the effect that the heat flux scales with ρu3 and is dominated by the density. From Table 4, ρu3 = 7.12 × 105 kJ/m2/s and 4.05 × 102 kJ/m2/s at the alti-tudes of 20 km and 90 km, respectively. The density decreases dramatically from 8.9 × 10−2 kg/m3 at the altitude of 20 km to

3.24 × 10−6 kg/m3 at 90 km; whereas, the velocity increases rela-tively moderately from 2.0 km/s at the altitude of 20 km to 5 km/s at 90 km. This temperature distribution is later used in the tem-perature jump boundary condition at the wall (Eq. (8)) during simulations of the complete 3D EXPERT model to calculate surface heat fluxes and the flow fields.

In the following discussion, flow analyses of the EXPERT cap-sule along the descent trajectory through the atmosphere are pre-sented. Fig. 12 shows the typical pressure and temperature con-tours on a cross-sectional plane of the EXPERT capsule at AOA =0 at an altitude of 40 km. The bow shockwave envelops the ve-hicle, while a secondary shock wave is formed at the flap-body junction. The Prandtl–Meyer expansion fan is associated with the sharp turn at the end-corner of the vehicle, and a recirculation zone is clearly observed in the vehicle’s wake. Near the recircula-tion region, the temperature increases at the vehicle base area and reaches the maximum near the rear stagnation point. These im-portant flow features were captured well by the current UNIC-UNS code. Notably, the flow characteristics are similar at different alti-tudes.

Figs. 13 and 14 show respectively the typical mass fraction dis-tributions of the neutral and ionized species of air surrounding the capsule at AOA = 0 at an altitude of 40 km. High amounts of dis-sociation and reaction are found in the front face of the EXPERT spacecraft due to the abrupt increase of temperature behind the shock wave and in the high-temperature wake region around the rear stagnation point. The most pronounced dissociation happens with respect to the oxygen molecules due to their corresponding low dissociation energy (Figs. 13(a) and 13(d)). On the other hand, the nitrogen molecules dissociate less significantly because of their high dissociation energy. Ionization effects are also notable in front of the capsule nose, particularly in the case of NO+ (Fig. 14(c)). This is caused by the shuffle reactions (O2 + N ! NO + O and N2 + O ! NO + N) and the dissociative-recombination reactions (N + O ! NO+ + e−) forming NO+ and free electrons (Fig. 14(f)), which, in terms of the overall composition of the gas, are small, but of particular significance [31].

H. Saldivar Massimi et al. / Aerospace Science and Technology 43 (2015) 360–371 369

Fig. 13. Mass fraction contours of the EXPERT capsule flow for an altitude of 40 km. (a) O2; (b) N2; (c) NO; (d) O; (e) N.

Fig. 14. Mass fraction contours of the EXPERT capsule flow for an altitude of 40 km. (a) O2+; (b) N2+; (c) NO+; (d) O+; (e) N+ and (f) electron number density of the EXPERT capsule flow for an altitude of 40 km.

370 H. Saldivar Massimi et al. / Aerospace Science and Technology 43 (2015) 360–371

Fig. 15. (a) Cp, (b) Cf and (c) Ch distributions over the EXPERT capsule at the altitude of 90 km, with no roll angle and an AOA of −5.5 .

Fig. 16. Comparison of (a) pressure coefficients, (b) skin friction coefficients and (c) Stanton numbers along the lower (windward) centerline of the EXPERT capsule for different altitudes.

Typical contour distributions of pressure, skin friction and heat transfer rate coefficients over the EXPERT capsule at the altitude of 90 km, with no roll angle and an AOA of −5.5 are demon-strated in Fig. 15. As shown, Cp, Cf and Ch are higher on the windward side (bottom half) than on the leeward side (upper half). The highest values of Cp and Ch occur in the nose stagnation re-gion, while that of Cf happens around the nose corner. The flow around the corner imposes a large favorable pressure gradient on the boundary layer, which results in an actual reduction in the boundary-layer thickness, causing the shear stresses to increase.

Fig. 16 shows the distributions of pressure, skin friction and heat transfer rate along the lower (windward) centerline of the EXPERT capsule surface at different altitudes, with no roll angle and an AOA of −5.5 . As seen, Cp, Cf and Ch distributions at differ-ent altitudes exhibit similar trends but with different magnitudes. Consistent with the findings in Fig. 15, the Cp and Ch distributions have their corresponding maxima located very close to the nose tip and Cf distribution has its maximum value around the nose corner. Cp, Cf and Ch decrease monotonically from their maximum-value points toward the flap. The oblique shock generated by the de-flected flap (see Fig. 12) causes abrupt increases of the values over the flap surface. The expansion fans at the rear edge of the flap yield sudden decreases in Cp, Cf and Ch. Abrupt decreases of Cp, Cf and Ch at X = 400 mm are caused by the expansion fans emit-ted at the ellipse-cone junction recess (Fig. 1). Notably, for cases where the altitude varies from 90 km down to 40 km, Cp, Cf and Ch distributions have higher values at higher altitudes. For the 20 km case, the trend is similar but the values of Cp, Cf and Chare higher than those at an altitude of 40 km due to the signifi-cant decrease in the freestream velocity (Fig. 2) and an increase in

the freestream density (Table 4). Nevertheless, the pressure, skin friction and heat flux actually increase as the EXPERT capsule de-scends. The increases in freestream density values as the altitude decreases from 90 km to 40 km result in the higher Cp, Cf and Chat higher altitudes.

6. Conclusion

The all-speed continuum code UNIC-UNS with finite-rate chem-istry and slip boundary conditions has been implemented to simu-late hypersonic flows in the transition regime over spherical-coned and EXPERT models. At first, the UNIC-UNS code and the bound-ary conditions adopted in present study were validated using the kinetic DSMC approach [32] and the works of Wen and Hornung [14]. For the cases of Knudsen smaller than 0.1, the results pre-dicted by the UNIC-UNS code were almost coincident with solu-tion obtained by the Boltzmann equation. Even when the Knudsen number reached 0.5, a good agreement of Stanton number, pres-sure and friction coefficients between UNIC-UNS code and the DSMC simulations were achieved in the vicinity of the body. It can be concluded that the UNIC-UNS code is reliable in simulat-ing hypersonic flows with rarefaction and thermochemical non-equilibrium effects. Next, the aerothermodynamic characteristics of the EXPERT capsule for the descent trajectory from 90 to 20 km were studied by the validated UNIC-UNS code. A detailed analysis of the aerothermodynamics for EXPERT was performed by utilizing the numerical flow visualization. The present numerical simula-tions provide some important data for EXPERT which cannot be easily derived by experiments. They provide useful numerical tools for the design and optimization of a space vehicle. Further analy-

H. Saldivar Massimi et al. / Aerospace Science and Technology 43 (2015) 360–371 371

ses of the entire descent trajectory of the EXPERT capsule will be conducted in the future so as to be able to compare with flight data once the EXPERT capsule has been launched.

Conflict of interest statement

We wish to confirm that there are no known conflicts of inter-est associated with this publication and there has been no signifi-cant financial support for this work that could have influenced its outcome.

Acknowledgements

The authors would like to thank the financial supports by the following funding: National Science Council of the Republic of China, Taiwan, under Contract No. NSC 100-2923-E-006-002-MY3. Research Grants Council, Hong Kong, under Contract No. GRF 526913 and The Hong Kong Polytechnic University under Depart-mental General Research Fund 4-ZZE8, by Department of Mechan-ical Engineering. The authors are also grateful to the National Cen-ter for High Performance Computing, Taiwan, for providing the computational resources.

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