Physics of High Speed Flows (I)

Contents 1 - Physics of High Speed Flows (I) ....................................................................................................... 2

1.1 - Atmospheric (RE-)Entry physics .................................................................................................. 2

1.2 - The ballistic coefficient and the trajectory and vehicle design..................................................... 4

1.3 - A typical example ........................................................................................................................ 12

1.4 - Major aerothermodynamic phenomena of re-entry vehicles .................................................... 13

1.5 - References ................................................................................................................................... 17

1 - PHYSICS OF HIGH SPEED FLOWS (I)

1.1 - Atmospheric (RE-)Entry physics

An important step forward for Space Exploration activities and for a more accurate knowledge of the Earth, and Universe is to develop the capability to send vehicles into space which select, collect and finally return samples from other celestial bodies to Earth where to perform their analysis. There are therefore, space missions that require the spacecraft (S/C) to go from space through the atmosphere to the surface (of Earth or another planet). This is currently called “Entry” and this process may be:

• re-entry, if the S/C is coming back to Earth, or

• entry, if the S/C is arriving and landing on another planet. In what follows we are going to refer to the re-entry process only.

Remark: There are also other types of “flights/situations” in which similar physical processes occur and this are the cases of the descent of ballistic missiles and/or of the sounding (suborbital) rockets.

In each case, there are the following main aspects to look at (because the mission of the S/C have to be accomplished and this is “a must”):

• Deceleration. The S/C payload (including crew) and structure limit the maximum deceleration (measured in “g’s”) allowed. This deceleration must be: a) low enough to prevent damages or injuries to occur (for instance, the trained human payload can endure without special equipments loads of about +/- 12-g for short periods and the structure may be designed consequently) and b) high enough to brake the S/C to (re-)enter rather than to skip off the atmosphere. There are some missions designed so that the deceleration must be realized without any active devices (like rocket motors and/or parachutes).

• Heating. The deceleration is associated with an intense heating caused by the heat exchange between the S/C and the air molecules. The heat flux transferred to the S/C is so important that the heating of the S/C may damaged or even destroyed. The entry trajectory and de thermal protection system of the S/C must be designed so that to prevent the effects of the heating.

• Accuracy of landing (or impact). The S/C will be expecting to arrive at an imposed location on the planet, with a given accuracy. This influences both the design of the S/C and entry trajectory.

The deceleration, heating and accuracy requirements define a permissible (re-)entry corridor, see Figure 2.1. If the S/C does not enter such a corridor, then it will experience either too much heating and burn or too little drag and skip off the atmosphere. The size of the corridor depends on three factors, which are: deceleration, heating and (landing) accuracy.

Figure 2.1 A typical reentry corridor for a S/C.

The space vehicle configuration and the entry trajectory are the result of a trade-off process, see Figure 2.2, which is based on the mission requirements for the S/C. Mission requirements affect directly the vehicle design.

Figure 2.2 The mission requirements determine the re-entry design process.

Remark. A high speed Earth re-entry vehicle has the following characteristics:

• entry velocity higher or equal to 11.7 km/s (compared to 7.5 km/s for the Space Shuttle),

• very high heat fluxes (more than 10 MW/m2) and • heat loads (in the range of 500 MJ/m2), where the radiative part is important.

The design of such a S/C relies on a good understanding of the aerodynamic loads encountered during the atmospheric part of the re-entry. The shape of such a capsule is the result of a trade-off design among hypersonic aerodynamic loads (aeroheating), subsonic drag and subsonic stability.

1.2 - The ballistic coefficient and the trajectory and vehicle design

The forces acting on the S/C are the weight and the aerodynamic drag and lift, see Figure 2.3. During the initial phase of the entry, one may assume that the aerodynamic drag is dominant. The flight-path angle, 𝛾, is the angle between the local horizontal and the velocity vector. The entry trajectory is characterized by the initial velocity and the entry flight-path angle, while the S/C shape and size and the thermal protection systems are characterizing the vehicle design.

If the drag force is given by an equation like 𝐷 = 12𝜌𝑉2𝑆𝐶𝐷, where the drag coefficient CD

depends on the vehicle shape and flying conditions and S is the (cross) reference area, then the acceleration on the trajectory is given by:

𝑎𝐷 =12𝜌𝑉

2

𝑚𝑆𝐶𝐷

. 2.1)

The ballistic coefficient is defined as the denominator of the equation (2.1),

𝐵𝐶 =𝑚𝑆𝐶𝐷

. 2.2)

Figure 2.3 The forces acting on the spacecraft.

From equation (2.1) one sees that the magnitude of the deceleration from drag is inversely related to BC. The BC quantifies an S/C mass, drag coefficient and (cross-sectional) area and predicts how drag will affect it. The Figure 2.4 shows an intuitive relation between the shape of the vehicle and its BC. As a result, the deceleration depends on BC and an S/C with small ballistic coefficient (a blunt body) will decelerate much stronger than one with a high BC (a streamlined body).

Figure 2.4 The S/C shape and its ballistic coefficient.

The trajectory design is based on controlling the re-entry initial conditions, i.e. the velocity and the flight path angle with which the S/C enters the atmosphere. For most missions, the re-entry initial conditions are set by the mission orbit and are difficult or even impossible to change significantly (without using propulsion).

The vehicle design is based on: a) choosing the S/C shape and size, see Figure 2.5, so that to have a convenient BC and after then, b) providing a thermal protection system to deal

with entry heating. The shape and size determine the magnitude of the aerodynamic forces and thus the ballistic coefficient.

Figure 2.5 Various S/C shapes and sizes and the corresponding BC’s.

The deceleration has a maximum value, that depends on the initial velocity and flight path angle, Figure 2.6. For a given re-entry flight-path angle, the higher the initial velocity, the greater the maximum acceleration. Further, for a given velocity, the higher the re-entry flight-path angle, the greater the maximum acceleration. The acceleration and its maximum value determine the time that an entering S/C takes to get down to dense layers of the atmosphere.

Using the flight dynamics equations one can find the S/C’s maximum deceleration and the altitude at which it occurs. These are given by:

𝑎𝑚𝑎𝑥 =𝑉𝑟𝑒−𝑒𝑛𝑡𝑟𝑦2 𝛽 sin𝛾

2 𝑒, where

𝛽 = 0.000139 𝑚−1(atmospheric scale height) and 𝑒 = 2.7182 (base of the natural logarithm).

2.3)

and

𝐻𝑎𝑚𝑎𝑥 = 1𝛽

ln � 𝜌0𝐵𝐶 𝛽 sin𝛾

�, where

𝜌0 = 1.225 𝑘𝑔/𝑚3(air density at the sea level).

2.4)

The maximum deceleration depends on the initial re-entry velocity and flight-path angle. However, the altitude of the maximum deceleration depends only on the flight-path angle, as can be noticed from (2.4).

Figure 2.6 Deceleration profiles for various re-entry velocities and flight-path angles.

Because of the extremely high re-entry velocities, the travel through the upper atmosphere produces significant aero-thermodynamic effects and these are related to the occurrence of the very strong shock waves, see Figure 2.7. The hot air downstream of the shock wave transfers some of its heat to the S/C by convection. The convection is known to be the primary means of heat transfer to a S/C entering the Earth’s atmosphere at speeds under about 15000 m/s. Above this speed, the air molecules are so hot that they transfer more of their energy to the S/C by radiation. The heat transfer process is quantified through the heating rate (heat energy per unit area per unit tome), ��, measured in watts per square meter.

For streamlined bodies the shock wave attaches to the tip and thus a significant amount of heat transfers to the body in this region, see Figure 2.8. These may cause a strong

localized heating of the S/C’s sharp tip and thus a very high temperature may be reached. Furthermore, the air flow near the surface does not restrain the heat transfer and as a result, the overall heating rate is high, as it is shown in Figure 2.9 (streamlined vehicles have a large

BC).

Figure 2.7 Attached and detached shock waves,

for high (streamlined) and low (blunt) BC vehicles.

If the vehicle is blunt, the shock wave detaches and curves in front of the vehicle, leaving a boundary of air between the shock wave and the S/C’s surface. This spreads the heat over a larger volume of air. The air flow near the the surface tends to slow down the convective heat transfer. Thus, the heating rate for blunt vehicles is relatively low.

Figure 2.8 The impact of the shape on the heating process.

Figure 2.9 shows how different BC affect the maximum heating rate. It is a matter of evidence that the heating rate is more severe and occurs much lower in the atmosphere for high BC vehicles (i.e., for streamlined S/C with sharp tip). This is due to the shape of the shock wave, as it was explained previously.

Figure 2.9 The dependence of the heating rate on the BC.

The heating rate is a function of the S/C’s velocity and nose radius (𝑟𝑛𝑜𝑠𝑒) and also to the density of the atmosphere. An empirical formula for the heating rate shows that:

𝑞 = 1.83 × 10−4𝑉3�𝜌

𝑟𝑛𝑜𝑠𝑒

.

2.5)

The equation (2.1) confirms that the smaller the nose radius, the higher the heating rate.

Figure 2.10Figure 2.10 shows a typical variation of the heating rate with the altitude and also the dependency of the peak heating rate value on the re-entry velocity and flight-path angle.

Figure 2.10 Variation of the heating rate with altitude,

at different re-entry velocities (left) and flight-path angles (right).

Using some algebraic manipulations one may find the velocity and the altitude where the maximum heating rate occurs. Thus, this velocity is given by

𝑉��𝑚𝑎𝑥 = 0.846 ∙ 𝑉𝑟𝑒−𝑒𝑛𝑡𝑟𝑦, 2.6)

and the corresponding altitude is

𝐻��𝑚𝑎𝑥 = 1𝛽

ln � 𝜌03𝐵𝐶𝛽 sin𝛾

�, where

𝛽 = 0.000139 𝑚−1(atmospheric scale height) and

𝜌0 = 1.225 𝑘𝑔/𝑚3(density at the sea level).

2.7)

From Figure 2.10 and the previous equations results that:

• the maximum heating rate increases as the re-entry velocity increases, • the velocity for the maximum heating rate is about 85% of the re-entry velocity, • the steeper the re-entry angle, the higher the maximum heating rate.

A steep re-entry angle causes a very high heating rate but for a short time, so the overall effect on the vehicle may be small. A low re-entry angle leads to much lower heating rates, but acting for longer times. The total heat load, Q, is the amount of thermal energy per unit surface (J/m2) the S/C receives and is obtained by integrating the heating rate over the entire re-entry time.

Figure 2.11 The total heat load for various re-entry velocities.

Q varies therefore with the velocity but it does not vary with the re-entry flight-path angle (the heat results from the mechanical energy dissipation process during re-entry). Figure 2.11 shows an example from which one may observe that the higher the re-entry velocity, the higher the total heat load is. Although the peak heating rate depends on the flight-path angle, the total heat load is constant for a given re-entry velocity.

Table 2.1. Ballistic Coefficient (BC) trade-offs for the re-entry

In what is concerning the accuracy, it must be noticed that the aerodynamic drag and lift forces perturb the descending trajectory from the path it would follow under gravity alone. The modeling of these forces (including the density of high atmosphere and the dimensionless coefficients) can be done only with approximations. To reduce these atmospheric effects on the accuracy, the trajectory must be so that the S/C spends the least time in the atmosphere. This leads to high re-entry velocities and steep re-entry flight-path angles and clearly, this choice increases the severity of the deceleration and heating rate.

The effects of the BC on the re-entry process are summarized in Table 2.1.

The important heat accumulation that occurs during the re-entry is counteracted by thermal protection systems. These are specially materials and design technique and include:

• Heat sinks (the heat is spread out and stored in the S/C, using an extra (quantity of) material to absorb the heat, thus keeping the peak temperature lower);

• Ablation (the re-entry vehicle is sheltered with a material having a very high latent heat of fusion, which absorbs through melting and/or vaporization a large amount of heat energy and disappears in atmosphere);

• Radiative cooling (the re-entry vehicle is sheltered with a material that has a high emissivity and thus it reaches the thermal equilibrium at a relatively low temperature by emitting back to the atmosphere almost as much heat energy as it absorbs).

The heat sinks are simple but heavy and they reduce the payload mass significantly. The ablation process can be used only once, because the S/C must be completely renewed for each mission. The radiative cooling requires a surface coating with high emissivity and melting point

and also an inner efficient insulator to protect the structure, but is the only solution used for reusable vehicles (the Space Shuttle, for instance).

The BC, the shape and size of the S/C, the parameters of the re-entry trajectory and the thermal protection system represent the factors that determine the physical processes that take place during the re-entry. Correspondingly, they also determine the mathematical models that may/must be used for the accurate and efficient quantitative prediction of the flow and heat transfer by means of numerical computation.

Remark on lifting re-entry. The use of the lift force offers more flexibility in the re-entry trajectory design and also opens the possibility of aerobraking, which is an exciting application mainly for the entry in the atmospheres of another planets. However, this does not change the facts presented previously. In other words, the braking in the atmosphere, the heating and the need for thermal protection system still remain the essential factors of the re-entry process.

1.3 - A typical example

We consider in what follows a typical flight scenario for a sample return mission. The re-entry trajectory is high speed one, with a steep descent characterized by an initial flight path angle of about -12.5 deg. The flight scenario is shown in Figure 2.12. Such a trajectory example:

• allows predicting the flow field around the proposed capsule because of it provides freestream conditions for CFD computations,

• defines the aerothermodynamic environment the capsule has to withstand during descent. For instance, the evolutions of heat fluxes (convective and radiative part) vs. the velocity are presented.

In the preliminary design the convective and radiative heat fluxes have been estimated by using analytical engineering correlations such as Scott relationship for convective heat flux and Tauber-Sutton one for the radiative one. These estimations are important for designing of the S/C heat shield, because the aeroheating environment dictates, in fact, the type and size of the thermal protection system to use.

Figure 2.12 Altitude vs. velocity and heating flux evolution for a typical sample return mission.

The peak heat rate generally determines the range of possible thermal protection material while the integrated heat load determines the thickness and hence the mass of the heat shield. Furthermore, the strong flowfield radiation and heatshield ablation determine the return vehicle’s aerothermodynamic performances.

Due to the very high temperatures reached in the shock layer caused by the strong bow shock in front of the capsule, the gas dissociates and ionized and also emits radiated energy. This energy travels through the flowfield and interacts with the gas itself. At the entry velocity predicted for this sample return mission this contribution, that is generally very low with respect to other “energies” in the flowfield, cannot be neglected because it can cause an additional source of heating load at the wall to be taken into account. The flowfield and the radiative heat field are coupled. Mathematically, this means that in the Navier-Stokes equations a source term is added to the energy equation to take into account the energy radiated from and towards the vehicle walls.

1.4 - Major aerothermodynamic phenomena of re-entry vehicles

Aerothermodynamics, or hypersonic aerodynamics in the frame of this work, is based on mathematical models that take into account the specific physical processes. When modeling these physical processes it is necessary to specify the class of the vehicles on which the focus is. Although there are numerous common aerothermodynamic features to all the hypersonic vehicles there are also very different key technology demands for different vehicle classes. In the frame of the present work we discern three classes of hypersonic vehicles:

• winged return vehicles (RV), like the Space Shuttle, BURAN and HERMES, • ascent and re-entry vehicles (ARV) and • aero-assisted orbital transfer vehicles (AOTV).

Each of these three classes has special aerothermodynamic features, which lead to different research, mathematical modeling and development needs. The features are presented in Table 2.2.

From the classification results that the following physical effects play a major role:

• viscosity effects, notably laminar-turbulent transition and turbulence, • thermodynamic effects, especially plasma (ionization, radiation), • heat loads and the cooling concept.

A review of the modeling and simulation means of the potential critically aerothermodynamic phenomena is presented in Table 2.3.

Table 2.3 shows that there are improvements to be done for the modeling and simulation of all the enumerated physical phenomena. The simulation accuracy demand for each phenomena depends on the sensitivity of the vehicle or its components on the respective phenomena.

Table 2.2. Comparative consideration of the aerothermodynamic features of the major classes of re-entry vehicles

RV ARV AOTV

Mach number range 28-0 0/7-28 20-35

Configuration blunt Opposite demands at ascent and re-entry (sharp/blunt)

blunt

Flight time through atmosphere

short short short

Angle of attack large small/large head on

Drag large small/large large

Lift to drag ratio small but larger than the others

small small

Thermal heating problems

loads loads loads

Flowfield pressure field dominated

viscous effects dominated / pressure field dominated

pressure field dominated

Rarefaction effects initially strong weak / initially strong strong

Thermodynamic effects

strong medium/strong strong

Critical components control surfaces, landing syst.

nozzle/base, control surfaces

control devices

Special problems large Mach number span

Propulsion integration, opposite demands

Plasma effects

Table 2.3. Review of the modeling and simulation means for aerothermodynamic phenomena Phenomena Modelling and simulation

means Vehicle class

Transition laminar-turbulent poor RV, ARV

Attached turbulent flow good RV, ARV, AOTV

Laminar strong interaction good RV, ARV, AOTV

Turbulent strong interaction fair RV, ARV

Laminar separation good RV, AOTV

Turbulent separation fair RV, ARV

Hypersonic viscous interaction (low density effects)

good RV, ARV, AOTV

Equilibrium real-gas effects good RV, ARV, AOTV

Non-equilibrium real-gas effects

good RV, ARV, AOTV

Turbulent heat transfer poor RV, ARV

Surface radiation cooling Good (in USA) RV, ARV, AOTV

Plasma effects fair AOTV

1.5 - References

1. Sellers, J. and all, Understanding Space: An Introduction to Astronautics, McGraw-Hill, 2000. 2. Aerothermodynamic Field Past a Reentry Capsule for sample Return Missions, by A. Viviani,

G. Pezzella, C. Golia, 28TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES, ICAS 2012.

3. Hypersonic Aerodynamics, by E.H. Hirschel, Deutsche Aerospace AG, Space Course 1993, TU Munich, Oct. 11-22, 1993.

4. Stagnation-Point Radiative Heating Relations for Earth and Mars Entries, by M.E. Tauber, K. Suttont, Journal of Spacecraft and Rockets, VOL. 28, NO. 1 (pag.40-42).

5. Thermal Design of Aeroassisted Orbital Transfer Vehicles, by C.D. Scott, R.C. Ried, C.P. Li and S.M. Derry, "An AOTV Aeroheating and Thermal Protection Study," H. F. Nelson (ed.), Thermal, Vol. 96 of Progress in Astronautics and Aeronautics, AIAA, New York, 1985, pp. 198-229.