- Email: [email protected]
Mars transportation vehicle concept
Acta Astronautica 103 (2014) 250–256
Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Mars transportation vehicle concept Maria Smirnova a,b,n a b
Moscow M.V., Lomonosov State University, Leninskie Gory, 1, Moscow 119992, Russia Saint Petersburg State Polytechnical University, 29 Politechnicheskaya Str., 195251 St. Petersburg, Russia
a r t i c l e in f o
abstract
Article history: Received 20 May 2014 Accepted 9 June 2014 Available online 24 June 2014
A concept of Mars planet propulsion vehicles is analyzed. Aluminum or magnesium combustion in CO2 is considered as the main energy production cycle. The flight possibilities in rarefied Martian atmosphere are analyzed. The problem of lift force determination in compressible gas in the proximity of rigid surface was solved theoretically. It was demonstrated that lift force increase on approaching rigid surface could guarantee reliable flights in Martian atmosphere. & 2014 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Martian atmosphere Combustion Metal powder Lift force Screen effect Screen-plane vehicle
1. Introduction The forthcoming Mars exploration missions to be effective by covering maximal territory in course of one mission will need a reliable transportation vehicle. This brings the problem of local resource utilization on top of current research needs. The concept of Martian resources utilization for transportation enables technology for exploration of Mars, which can significantly reduce the mass, cost, and risk of robotic and human missions. The problem of rover versus hopper is to be solved. Both approaches have their advantages and disadvantages. In this paper we will concentrate on airborne propulsion. The critical element in future missions is the large mass of propellant for a Mars ascent vehicle, as well as power and propellant to accommodate a long stay and mobility on Mars. Transportation of propellant from Earth to Mars requires a great increase in the initial mass of hardware in low Earth orbit. Most promising approach in the Martian resources utilization concept suggests using the Martian CO2 directly
n
Tel./fax: þ7 495 9393754. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.actaastro.2014.06.015 0094-5765/& 2014 IAA. Published by Elsevier Ltd. All rights reserved.
as an oxidizer in a jet or rocket engine [1], because Martian atmosphere contains 95.3% carbon dioxide. This approach is based on the unique ability of some metals and compounds to burn with CO2. The idea to burn metals in atmospheres of planets for propulsion appeared long ago. In early studies by Yuasa and Isoda [2], “CO2-breathing” jet engines were considered. This looks advantageous, because no CO2 processing is required and the propulsion system is the only element that needs to be developed in this case. For human missions, however, rocket engines are required. The low temperature of the Mars atmosphere favors CO2 liquefaction, thus allowing relatively easy accumulation of liquid CO2 at pressure 10 bar for subsequent use in rocket engines [1]. On the other hand, the low atmospheric pressure of Mars (7–9 mbar), being a major problem for jet engines, is perfect for rockets (no need for pressure higher than 10 bar in the combustion chamber to obtain high nozzle expansion ratio). These favorable circumstances were noted by Shafirovich et al. [3], who first conducted thermodynamic calculations of performance characteristics for rocket engines using metal-CO2 propellants and made ballistic estimates. Then successful design solutions were developed for engines using powdered metal fuel with air or steam, which could be used in metal-CO2
M. Smirnova / Acta Astronautica 103 (2014) 250–256
propulsion systems. Fundamental studies of metals combustion have improved significantly understanding of Mg and Al particle combustion in CO2 atmosphere. Progress was made in development of methods for liquid CO2 production on Mars, and new ideas appeared for production of metal fuel on Mars. Finally, mission analyses have identified scenarios in which metal-CO2 propulsion promises great advantages. Performance characteristics of ramjet engines in the Martian (CO2) atmosphere were calculated for use of magnesium [2] as fuel. Unfortunately, low atmospheric pressure on Mars leads to either low thrust, or large specific fuel consumption and large inlet and exhaust nozzle sizes [2]. Turbojet could be more efficient, but presence of oxide particles in exhaust gases could lead to particulate phase deposition and damaging the turbine blades. Pulse detonation engines [4,5] could be used much more successfully as they do not need turbines and provide higher specific impulse. However, periodical detonation onset becomes important [6]. Besides, using metal as fuel and CO2 as an oxidant brings the problem of detonation onset and control in such a two-phase system. The equilibrium calculation and experimental study on the ignition and combustion characteristics of magnesium, lithium, aluminum and boron in a CO2 atmosphere suggest that magnesium is the most attractive fuel for the CO2breathing engine using in Mars atmosphere because of its easy ignitability and fairly fast burning rate [2], despite of the fact its specific energy release in CO2 is twice less than in O2 combustion, and 3–4 times less than hydrocarbon fuels burning in oxygen. Another problem relevant to flights in Martian atmosphere is that of low density being approximately 60 times less than atmosphere on Earth, which means the Mars surface density is equal to atmospheric density on Earth at the altitude of 15 km. The lift force being directly proportional to atmospheric density provides much less favorable flight conditions. Combining this factor with lower thrust characteristics provided by metal burning in CO2 engines, probably, brought the concept of rover dominating over hopper. The aim of the present paper is reconsidering traditional flight concept for Mars applications.
2. Aerodynamic screen effect Making use of classical airplane flight concept in Martian atmosphere could be hardly realized due to atmospheric density being 60 times less than on Earth. The fact that gravity on Mars surface is only 38% of that on Earth does not compensate the difficulty, because for a typical air-plane weight being decreased 2.6 times the lift force should decrease 60 times for one and the same cruising velocity. The classical aerodynamics concept says that the lift force for subsonic wings is generated due to pressure differential on both sides of wing in streaming flow. The classical formulas for lift force of a single wing do not take into account the presence of other bodies in the flow. However, it is known that wing moving near rigid boundary produces strong effect on the forces acting on wing from streaming fluid [7].
251
At the beginning of the XX century it was observed that the lift force of a wing moving near flat surface increases strongly in comparison with free flight. An article about screen effect by Juriev [8] was published in 1923 in the USSR. That fact was used in creation of new flying devices – screen-planes, which got the Russian name “ekranoplan”. In 1932 Grohovsky constructed a full-scaled model of a new marine flying device – catamaran. At the same time Finnish engineer T. Kaario proceeded to test his flying apparatus that used a screen effect. Then (1963–1976) a Soviet machinery designer R.L. Bartini created a screenplane project SVVP-2500 that took off in 1974. The first Soviet manned jet screen-plane SM-1 was created in collaboration with R. Alekseev in 1960–1961 [9]. Giant screen-plane KM was constructed by 1966 and “Orlyonok” type screen-planes were built from 1974 to 1983.[7] Designing of new flying devices continues in many countries. Sedov obtained an analytical solution for the lift force in terms of Weierstrass functions [10] using theory of a complex variable. Approximate analytical solution of the problem of non-steady plane moving near rigid surface was obtained by Rozjdestvensky [11] using asymptotic expansion. Theoretical investigation of a wing moving near rigid surface was made by Panchenkov [12,13] and Gorelov [14], but the obtained solutions incorporated free constants. Experimental results are shown in [15]. Numericoanalytical solutions were obtained in [16]. Below numerical solution and analytical formulas for the wing flying near rigid surface will be obtained and analyzed for being applied in Martian flight vehicle design.
3. Mathematical problem statement We regard plane wing motion in compressible ideal fluid. The system of equations for fluid flow in the motionless co-ordinates is the Euler system ∂ρ ∂ρ ∂ρ ∂V x ∂V y þV x 0 þ V y 0 þ ρ þ 0 ¼ 0; 0 ∂t ∂x ∂y ∂x ∂y ∂V x ∂V x ∂V x 1 ∂p þV x 0 þV y 0 ¼ ; ∂t ∂x ∂y ρ ∂x0 ∂V y ∂V y ∂V y 1 ∂p þVx 0 þVy 0 ¼ : ∂t ∂x ∂y ρ ∂y0
ð1Þ
k Flow is assumed adiabatic p=p0 ¼ ρ=ρ0 . In case of steady-state flight conditions at a constant altitude h above the surface the system of equations could be regarded in the moving co-ordinate system x ¼ x0 þ Vt; y ¼ y0 (V is the parallel to surface flight velocity), where it takes the stationary form. We assume disturbances being small and attack angle (α) being small as well, thus neglecting the second order of magnitude terms in linearization of system (1), which yields 1 ∂p ∂V x ∂V y þ ρ0 þ 0 ¼ 0; p ¼ p0 þ a2 ρ ρ0 ; 0 2 ∂x ∂y a ∂t ∂V y dp ∂V x 1 ∂p 1 ∂p ¼ ¼ a2 ¼ ; : ð2Þ dρ ρ ¼ ρ 0 ∂t ρ0 ∂x0 ∂t ρ0 ∂y0
252
M. Smirnova / Acta Astronautica 103 (2014) 250–256
For small disturbances system (2) testifies that acceleration vector has potential ∂u ∂ p ∂v ∂ p ¼ 0 ¼ 0 ; ∂t ∂x ρ0 ∂t ∂y ρ0 which yields velocity potential Z t p ∂φ ∂φ φ¼ dt; u ¼ 0 ; v ¼ 0 ∂x ∂y t 0 ρ0
ð3Þ
Boundary conditions for the system of equations include slip conditions on the wing surface and on the ground surface. After transfer to another independent variable pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x~ ¼ x=δ; δ ¼ 1 M 2 the equations and boundary conditions describing the gas flow take the following form:
0 r x~ r L=δ;
p ¼ p0
∂φ ~ ¼ V α 7 ðδxÞ; ∂y ∂φ y ¼ 0; ¼0 ∂y 7
y¼h ;
1 o x~ o 1;
ρ0 V ∂φ : δ ∂x~
ð5Þ
Boundary conditions should be supplemented with function behavior at the infinity for the uniqueness of the solution. Thus an analytical function satisfying boundary conditions and decreasing at the infinity should be developed. 4. Analytical solution and approximate formulas The solution of the Laplace equation can be developed in the form of a real part for the analytical function of ~ yÞ ¼ Re ΦðzÞ; z ¼ x~ þ iy. Actually, a complex variable φðx; it is necessary to develop first derivative of the analytical 0 function, which could be denoted as: iTðzÞ ¼ Φ ðzÞ. The development of the analytical function is reduced to the following boundary problem: 0 r x~ r L=δ;
7
y¼h ;
1 o x~ o 1; y ¼ 0;
~ ¼ V α~ 7 ðxÞ ~ ReTðx~ þ ihÞ ¼ V α 7 ðδxÞ
~ ¼0 ReTðxÞ
The pressure can be developed from the following formula: pðx~ þ iyÞ ¼ p0 þ
0 r x~~ r 1;
~7 y~~ ¼ h~ ;
1o x~~ o1; y~~ ¼ 0;
Thus gas flow with small disturbances in potential field of mass forces is always potential. The solution is equivalent to solution of the system 2 1 ∂2 φ ∂ φ ∂2 φ ∂φ ð4Þ þ 02 ¼ 0; p ¼ p0 ρ0 : 2 2 02 ∂t a ∂t ∂x ∂y
∂2 φ ∂2 φ þ ¼ 0; ∂x~ 2 ∂y2
The development of the analytical function is reduced to the following boundary problem:
ρ0 V ImTðx~ þiyÞ δ
For the uniqueness of the solution boundary conditions should be supplemented with function behavior at the infinity and at the ends of the segment 0 r x~ rL=δ. It is considered that the function tends to zero at the infinity and is limited at the rear edge of the wing, which is the result of the Chaplygin–Zhukovsky hypothesis. The solution is developed with the help of the symmetry principle. The following non-dimensional variables ~ ~ ~ L; are introduced x~~ ¼ x= y~~ ¼ y=L; h~ ¼ h=L; L~ ¼ L=δ.
~~ ~~ ¼ V α~~ 7 ðxÞ ~~ Re Tðx~~ þ ihÞ ¼ V α 7 ðLxÞ
~~ 0 þ Þ ¼ 0 Re T þ ðx;
The tildes will be omitted in the following equations. The solution of the problem can be derived in the form of Cauchy type integral [17]: " # Z 1 Z 1 1 1 μðtÞ 1 ζ ðtÞ þ iηðtÞ dt þ dt TðzÞ ¼ Χ 1 ðzÞ 2π i 0 Χ 1þ ðtÞ t þ ih z 2π i 0 t þ ih z " # Z 1 Z 1 1 1 μðtÞ 1 ζ ðtÞ iηðtÞ dt þ dt Χ 2 ðzÞ þ 2π i 0 Χ 2 ðtÞ t ih z 2π i 0 t ih z where
qffiffiffiffiffiffiffiffiffi z z1 z1 ¼ 1 þ ih; z2 ¼ ih z z2 ; ffiffiffiffiffiffiffiffiffiffiffi ffi s z z1 Χ 2 ðzÞ ¼ ; z1 ¼ 1 ih; z2 ¼ ih z z2
Χ 1 ðzÞ ¼
The function TðzÞ satisfies the boundary condition 1 o x o 1; y ¼ 0; Re T þ ðx; 0 þ Þ ¼ 0. þ It is assumed that μðsÞ ¼ Vðα~~ ðsÞ þ α~~ ðsÞÞ; ζ ðsÞ ¼ þ ~ ~ ~ ~ Vðα ðsÞ α ðsÞÞ. The function TðzÞ will satisfy the 7 boundary conditions 0 rx r 1; y ¼ h ; Re Tðx þ ihÞ ¼ 7 ~ V α~ ðxÞ if the following singular integral equation is fulfilled: Z 1 Z 1 1 ηðtÞ 1 ðt xÞηðtÞ dt dt 2π 0 t x 2π 0 ðt xÞ2 þ 4h2 Z 1 ~þ Vh α~ ðtÞ α~~ ðtÞ dt π 0 ðt xÞ2 þ 4h2 r Z 1 ffiffiffiffiffiffiffiffiffiffi þ V t ðt xÞ ðα~~ ðtÞ þ α~~ ðtÞÞdt RðxÞ sin αðxÞ 2π 0 1 t ðt xÞ2 þ 4h2 Z rffiffiffiffiffiffiffiffiffiffi þ Vh 1 t 1 þRðxÞ cos αðxÞ ðα~~ ðtÞ π 0 1 t ðt xÞ2 þ 4h2 þ α~~ ðtÞÞdt ¼ 0: where 1 2h 2h þ arctg ; αðsÞ ¼ arctg 2 1s s
RðsÞ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 ð1 sÞ þ4h 2
s2 þ4h
:
7 If the wing has a shape of a plate α~~ ¼ γ the singular integral equation takes the following form: Z Z 1 1 ηðtÞ 1 1 ðt sÞηðtÞ dt dt π 0 t s π 0 ðt sÞ2 þ 4h2 Z 1 rffiffiffiffiffiffiffiffiffiffi Vγ t ðt sÞ sin αðsÞRðsÞ dt þ 1 t ðt sÞ2 þ 4h2 π 0 r ffiffiffiffiffiffiffiffiffi ffi Z 1 2Vhγ t dt cos αðsÞRðsÞ ¼0 1 t ðt sÞ2 þ 4h2 π 0
The third and the fourth integrals can be taken analytically using the theory of residues. The following singular integral equation can be determined: Z Z 1 1 η~ ðtÞ 1 1 ðt sÞη~ ðtÞ dt ¼ dt FðsÞ π 0 t s π 0 ðt sÞ2 þ 4h~ 2
M. Smirnova / Acta Astronautica 103 (2014) 250–256
253
where
η~ ðsÞ ¼ ηðsÞ=V γ ;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 ð1 sÞ þ 4h ; FðsÞ ¼ RðsÞ sin αðsÞ 1; RðsÞ ¼ 2 2 s þ 4h 1 2h 2h þ arctg : αðsÞ ¼ arctg 2 1 s s
4.1. Determination of the lift force After the regularization the singular integral equation is reduced to the Fredholm equation: Z 1 η~ ðxÞ þ η~ ðtÞKðx; tÞdt ¼ GðxÞ; 0
Fig. 1. Schematic picture for a model wing profile simulated numerically.
where Kðx; tÞ ¼
1 4π
rffiffiffiffiffiffiffiffiffiffi 1x x
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin αðtÞ δðx; tÞ ; 2 2 RðtÞ ½tð1 xÞ xð1 tÞ þ 4h
rffiffiffiffiffiffiffiffiffiffi Z rffiffiffiffiffiffiffiffiffiffi 1x 1 1 s FðsÞds ; GðxÞ ¼ x π 0 1s sx
δðx; tÞ ¼ arg ½tð1 xÞ xð1 tÞ þ 2hi ; 1 2h 2h ; αðtÞ ¼ arctg þarctg 2 t 1t sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 ð1 tÞ þ 4h RðtÞ ¼ 2 t 2 þ 4h The following formula is used to determine the lift force: Z L Z L=δ
p ðsÞ p þ ðsÞ ds ¼ δ p ðs~ Þ p þ ðs~ Þ ds~ ð6Þ FΠ ¼ 0
0
The pressure is derived from the problem statement: pðs~ Þ ¼ p0 þ ρ0δV Im p ðs~ Þ p þ ðs~ Þ ¼
Tðs~ Þ
ρ0 V ImðT ðs~ Þ T þ ðs~ ÞÞ δ
In the case of a plate α~~ ¼ γ the lift force is determined by formula: " # Z i ρ π LV 2 γ 1 1h 1þ η~ ðs~~ Þ ds~~ ð7Þ FΠ ¼ 0
π
δ
0
If the wing has a model convex shape, for example þ α~~ ðxÞ ¼ 2γ x; α~~ ðxÞ ¼ γ , the lift force derived from (6) has the following form: "Z rffiffiffiffiffiffiffiffiffiffi Z rffiffiffiffiffiffiffiffiffiffiffi # Z 1 ρ0 V 2 γ L 1 1 s 1 1 t 0 2t 0 þ 1 FП ¼ η~ ðsÞds dt ds s π 0 δ 1 t0 t0 s 0 0 "
ρ V 2 γ Lπ 5 1 ¼ 0 4 π δ
Z
1 0
#
η~ ðsÞds
ð8Þ
The derived lift force differs from its analog for boundless surface by an extra summand, which decreases with the increase of the height. Formula (6) provides an estimate for the increase of the lift force on approaching the rigid surface, which is crucial for flights in Martian atmosphere.
5. Numerical solution. For the obtained result being verified the problem of lift force in streaming flow in the vicinity of rigid boundary was solved numerically. Numerical solution of the problem of potential streaming flow of incompressible fluid was performed using collocation method [18], which is based on distributing boundary panels with dipoles and a vortex at the tale end of the contour. The method was modified for solving problems in a bounded space using reflection techniques. The streamed contour ∂Ω is split by N þ1 grid points ξk , kK¼o,…,N so that the first and the last coincide ξo ¼ ξN at the tale end (Fig. 1). A pair of grid points [ξk , ξk þ 1 ] determines the panel segment ∂Ωk, the panel segments approximate entire wing profile. The velocity is given by the following formula: 2 3 N1 uPV ðx; ξk Þ uPV ðx; ξk Þ þ 4 5 uðxÞ ¼ U þ ∑ μk uPV ðx; ξk þ 1 Þ uPV ðx; ξk þ 1 Þ k¼0 h i þ κ uPV ðx; ξ0 Þ uPV ðx; ξ0 Þ is the velocity of undisturbed gas, ξ ¼ ! ! ξ r y ξ¼ ; uPV ¼ 1=2π r 2 is the veloη þr x
where U !
ξ ; η
city of disturbed gas from a vortex with the center in ξ and a unit power, r x ¼ x ξ;
r y ¼ y η;
r 2 ¼ r 2x þ r 2y ;
x ¼ ðx; yÞ
½ξk ; ξk þ 1 ; k ¼ 1:::N are the segments, which approximate the contour. Boundary condition on the rigid surface is satisfied. Unknown coefficients μk ; k can be developed from the streaming conditions on the contour, fulfilled in the centers of the segments. The system of linear algebraic equations is closed with the Chaplygin–Zhukovsky hypothesis, which minimizes the module of the velocity at the rear edge of the contour:
μ0 μN 1 þk ¼ 0
254
M. Smirnova / Acta Astronautica 103 (2014) 250–256
The lift force can be derived from the following equation: I 1 N 1 2 pn dσ ¼ ρ ∑ Δk nk u^ k ; F¼ 2 k¼0 ∂Ω where u^ k ¼ uðx^ k Þ;
x^ j ¼ 12 ðξj þ ξj þ 1 Þ
6. Results and discussion. The dependence of the reduced lift force F Π δ=ρ0 V 2 πγ L of a plate and a convex contour upon the altitude ~ H ¼ h~ ¼ hδ=L, which was developed analytically, is shown in Fig. 2. It can be seen that the lift force decreases with the increase of the height above the rigid surface until the force reaches its magnitude in a boundless medium. The lift force of a convex contour is bigger than that of a plate for the same altitudes. In Fig. 3. the dependence of the reduced lift force ~ F Π =ρ0 V 2 πγ L of a plate upon the altitude H ¼ h~ ¼ hδ=L, which was obtained analytically, is compared with the numerical solution for the Math number M ¼0.2 and for small attack angles less than 31. It is seen that for small attack angles the analytical and numerical solutions coincide, while for large angles surpassing 101 that would not be the case, and numerical solution should be used. Streamlines and the distribution of velocity along them obtained in numerical solution are shown in Fig. 4. for the case of gas flows streaming a plate with an inclination angle α moving at an altitude H ¼ h=L. The velocity module is presented on the scaling map in the right hand side of fig. 4, the measure unit is the velocity of undisturbed gas. It can be seen that the lift force of a plate with an inclination angle less than 301 decreases with the increase of the
Fig. 2. The dependence of the reduced lift force F Π δ=ρ0 V 2 πγL of a plate (lower curve) and a convex contour (upper curve) upon the altitude ~ H ¼ h~ ¼ hδ=L
Fig. 3. The dependence of the reduced lift force F Π =ρ0 V 2 πγL of a plate ~ upon the altitude H ¼ h~ ¼ hδ=L for M ¼ 0.2 obtained analytically (blue curve) and numerically (red curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
altitude above the surface. There is no such an effect for inclination angles more than 301. Analysis of the obtained results shows that a motionless surface (screen) essentially affects the lift force at the altitude, which is comparable with the length of a wing. If the altitude above the surface several times surpasses the wing span the screen effect practically disappears and the lift force tends to its value in an unbounded space. The effect of lift force increase on approaching the rigid surface manifests for relatively small attack angles (below 301). For small Mach numbers the lift force coincides with that determined by the classical Zhukovsky solution for incompressible fluid F П ¼ ρ0 V 2 πγ L. The increase of the flight Mach number brings to an increase of the lift force, as well as drag force. The screen effect still manifests in the same proportion. However, the increase of Mach number brings to growing linear theory error ( M2 ), thus on its approaching unity the obtained solution loses validity. Analysis of results presented in Fig. 2 enables to perform the following estimates. On approaching the wing to a plane screen at an altitude comparable with the wing length there appears a small screen effect manifesting in an increase of the lift force 3–4% as compared to that at high altitudes. The further decrease of altitude down to 0:1L brings to an increase of lift force 37–38% as compared to that at high altitudes (34% as compared with the altitude L). If the thickness of a wing is taken into consideration the lift force of a convex wing is bigger than that of a plate. Nevertheless, screen effect manifests for a convex wing as well (Fig. 4), but at a lower level. Estimates show, that for convex wing regarded in the present paper the increase of the lift force on decreasing altitude from L down to 0:1L is only 25%. However, the absolute value of the lift force is
M. Smirnova / Acta Astronautica 103 (2014) 250–256
255
Fig. 4. Streamlines and the distribution of velocity along them in case of gas flows around a plate with an inclination angle α on the altitude H ¼ h=L: (a) α ¼101, H¼ 0,1; (b) α ¼101, H¼ 0,5; (c) α ¼101, H¼ 1; (d) α ¼201, H¼0,5; (e) α ¼201, H¼ 1; (f) α¼ 301, H¼ 0,5; and (g) α¼ 301, H¼ 1;.
higher than that for a plane wing at an altitude 0:1L(surpasses 20%) thus providing the gain of 60% with the reference value. The manifestation of screen effect for convex wings essentially depends on the wing profile. The developed method makes it possible to evaluate screen effect for wings of different profiles. 7. Conceptual design of Mars vehicle Analyzing the Mars flight conditions as compared with that on Earth one can conclude that the conditions are less favorable. Despite of the fact that Mars gravity is 2.65 times less than that on Earth, and sonic speed in Martian atmosphere is 2 times less than that on Earth, which gives a gain in lift capacity for one and the same flight velocity,
the 60 times lower density brings to an overall loss. For example, for flying at a speed of 150 m/s the total loss in lift capacity will be 6 times. However, formulas (7), (8) and results illustrated in Fig. 3 show that the increase of wing span and decrease of flight altitude above the planet surface could compensate that loss and even bring to some gain in lift capacity. This flying vehicle utilizing the screen effect should rather have a discos convex shape. An artistic view of possible screen flight for Martian atmosphere is shown in Fig. 5. The two engines are located on top of the vehicle for avoiding dust sucking while flying near the planet surface. The discos shape provides the higher value for characteristic size parameter L. This shape is considered to be more effective; however the necessity of stabilization and flight control could need additional
256
M. Smirnova / Acta Astronautica 103 (2014) 250–256
First estimate for the shape of perspective vehicle made it possible recommending a convex shape with upper parabolic profile and lower linear profile. The principle is suggested for designing Martian screen-plane vehicles. Acknowledgment. The research was partially supported by Russian Foundation for Basic Research (Grant 12-08-91702). References.
Fig. 5. An artistic view of perspective screen-flight vehicle for Martian atmosphere.
Fig. 6. Screen-plane KM designed by R. Alekseev.
wings and/or tales. It is magnificent, that the cruising flight regime is stable in terms of flight altitude: on decreasing altitude the lift force increases (Fig. 2), and on incidental increase of altitude the lift force decreases thus returning vehicle back to the assigned altitude. The analogous flight vehicles were used on Earth for the purpose of increasing lift capacity in the territory of steppes, deserts and seas [7]. Fig. 6 illustrates one of vehicles used in the region of Caspian Sea. It is seen from the figure that wings have practically square shape, engines are located on top of the front and rear parts, and a big tail for maneuvering is necessary. 8. Conclusions The paper presents an analysis of flight options in Martian atmosphere. It is demonstrated, that using metal combustion rocket engines with carbon dioxide as an oxidizer makes it possible obtaining necessary thrust for flying in Martian atmosphere. Both analytical and numerical solutions determining lift force of a wing moving in the proximity of a rigid screen demonstrate the increase of the lift force on approaching screen, and provide formulas for estimating lift force for a flight vehicle moving in compressible gas near rigid surface.
[1] E. Shafirovich, A. Varma., Metal-CO2 propulsion for Mars missions: current status and opportunities, J. Propuls. Power 24 (3) (2008) 385–394. [2] S. Yuasa, H. Isoda, CO2-Breathing Propulsion for a Mars Airplane, AIAA Paper 89-2863, July 1989. [3] E. Ya. Shafirovich, A.A. Shiryaev, U.I. Goldshleger, Magnesium and carbon dioxide: a rocket propellant for Mars missions, J. Propuls. Power 9 (2) (1993) 197–203. [4] V.F. Nikitin, V.R. Dushin, Y.G. Phylippov, J.C. Legros, Pulse detonation engines: technical approaches, Acta Astronaut. 64 (2009) 281–287. [5] N.N. Smirnov, V.F. Nikitin, S. Alyari Shurekhdeli, Investigation of Selfsustaining waves in metastable systems: deflagration-to-detonation transition, J. Propuls. Power 25 (3) (2009) 593–608. [6] S.V. Khomik, B. Veyssiere, S.P. Medvedev, V. Montassier, G.L. Agafonov, M.V. Silnikov, On some conditions for detonation initiation downstream of a perforated plate, Shock Waves 23 (3) (2013) 207–211. [7] M.N. Smirnova, A.V. Zvyaguin, Theoretical solution for the lift force of ecranoplan moving near rigid surface, Acta Astronaut. 68 (11–12) (2011) 1676–1680. [8] B.N. Juriev, Earth Impact on the Aerodynamical Characteristics of a Wing, «Air fleet herald» №1, 1923. [9] N.I. Belavin, Screen-flights (Ecranoplan), Shipbuilding Publ., Moscow, 1977. [10] L.I. Sedov, Plane Problems of Hydrodynamics and Aerodynamics, Science Publ., Moscow, 1980. [11] K.V. Rozjdestvensky, Asymptotic theory of a wing moving on small distances from a rigid wall, Mech. Fluid Gas (№6) . [12] A.N. Panchenkov, Hydrodynamics of An Underwater Wing, Book Printing House № 5, Kiev, 1964. [13] A.N. Panchenkov, Theory of Optimal Supporting Surface, Science, Novosibirsk, 1983. [14] D.N. Gorelov, Methods of Solving Two-dimensional Boundary Problems in the Theory of a Wing, SO RAN, Novosibirsk, 2000, 215. [15] Y.М. Serebriisky, S.А. Biyachuev, The Study of Horizontal Steady Wing Motion in a Pipe, TSAGI papers (Central Aero Hydrodynamic Institute), issue 437, 1939. [16] K.V. Frolova, V.A. Frolov, Lift force of a plate near screen, in: Samara State Aerospace University Proceedings, 2009. [17] F.D. Gahov, Boundary Problems, Science, Moscow, 1977. [18] C. Pozrikidis, Fluid Dynamics: Theory, Computation and Numerical Simulation, Springer, New York, 2009. Maria Smirnova, born in Moscow in 5 November 1987. Graduated from Moscow M. V. Lomonosov State University in 2010 and got the degree of Master of Science in mechanics of fluids and gas. Submitted thesis in 2013 and got the Ph.D. degree in Physics and Mathematics. Research interests: fluid physics and mathematical physics. She is currently assistant professor in the Faculty of Mechanics and Mathematics, Moscow M.V. Lomonosov State University. At the same time holding half a position of visiting Associate Professor in Saint Petersburg State Polytechnical University. Received Scientific scholarships of O. Deripaska in 2010 and 2012, and scientific scholarship of the President of Russian Federation in 2013.
Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Mars transportation vehicle concept Maria Smirnova a,b,n a b
Moscow M.V., Lomonosov State University, Leninskie Gory, 1, Moscow 119992, Russia Saint Petersburg State Polytechnical University, 29 Politechnicheskaya Str., 195251 St. Petersburg, Russia
a r t i c l e in f o
abstract
Article history: Received 20 May 2014 Accepted 9 June 2014 Available online 24 June 2014
A concept of Mars planet propulsion vehicles is analyzed. Aluminum or magnesium combustion in CO2 is considered as the main energy production cycle. The flight possibilities in rarefied Martian atmosphere are analyzed. The problem of lift force determination in compressible gas in the proximity of rigid surface was solved theoretically. It was demonstrated that lift force increase on approaching rigid surface could guarantee reliable flights in Martian atmosphere. & 2014 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Martian atmosphere Combustion Metal powder Lift force Screen effect Screen-plane vehicle
1. Introduction The forthcoming Mars exploration missions to be effective by covering maximal territory in course of one mission will need a reliable transportation vehicle. This brings the problem of local resource utilization on top of current research needs. The concept of Martian resources utilization for transportation enables technology for exploration of Mars, which can significantly reduce the mass, cost, and risk of robotic and human missions. The problem of rover versus hopper is to be solved. Both approaches have their advantages and disadvantages. In this paper we will concentrate on airborne propulsion. The critical element in future missions is the large mass of propellant for a Mars ascent vehicle, as well as power and propellant to accommodate a long stay and mobility on Mars. Transportation of propellant from Earth to Mars requires a great increase in the initial mass of hardware in low Earth orbit. Most promising approach in the Martian resources utilization concept suggests using the Martian CO2 directly
n
Tel./fax: þ7 495 9393754. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.actaastro.2014.06.015 0094-5765/& 2014 IAA. Published by Elsevier Ltd. All rights reserved.
as an oxidizer in a jet or rocket engine [1], because Martian atmosphere contains 95.3% carbon dioxide. This approach is based on the unique ability of some metals and compounds to burn with CO2. The idea to burn metals in atmospheres of planets for propulsion appeared long ago. In early studies by Yuasa and Isoda [2], “CO2-breathing” jet engines were considered. This looks advantageous, because no CO2 processing is required and the propulsion system is the only element that needs to be developed in this case. For human missions, however, rocket engines are required. The low temperature of the Mars atmosphere favors CO2 liquefaction, thus allowing relatively easy accumulation of liquid CO2 at pressure 10 bar for subsequent use in rocket engines [1]. On the other hand, the low atmospheric pressure of Mars (7–9 mbar), being a major problem for jet engines, is perfect for rockets (no need for pressure higher than 10 bar in the combustion chamber to obtain high nozzle expansion ratio). These favorable circumstances were noted by Shafirovich et al. [3], who first conducted thermodynamic calculations of performance characteristics for rocket engines using metal-CO2 propellants and made ballistic estimates. Then successful design solutions were developed for engines using powdered metal fuel with air or steam, which could be used in metal-CO2
M. Smirnova / Acta Astronautica 103 (2014) 250–256
propulsion systems. Fundamental studies of metals combustion have improved significantly understanding of Mg and Al particle combustion in CO2 atmosphere. Progress was made in development of methods for liquid CO2 production on Mars, and new ideas appeared for production of metal fuel on Mars. Finally, mission analyses have identified scenarios in which metal-CO2 propulsion promises great advantages. Performance characteristics of ramjet engines in the Martian (CO2) atmosphere were calculated for use of magnesium [2] as fuel. Unfortunately, low atmospheric pressure on Mars leads to either low thrust, or large specific fuel consumption and large inlet and exhaust nozzle sizes [2]. Turbojet could be more efficient, but presence of oxide particles in exhaust gases could lead to particulate phase deposition and damaging the turbine blades. Pulse detonation engines [4,5] could be used much more successfully as they do not need turbines and provide higher specific impulse. However, periodical detonation onset becomes important [6]. Besides, using metal as fuel and CO2 as an oxidant brings the problem of detonation onset and control in such a two-phase system. The equilibrium calculation and experimental study on the ignition and combustion characteristics of magnesium, lithium, aluminum and boron in a CO2 atmosphere suggest that magnesium is the most attractive fuel for the CO2breathing engine using in Mars atmosphere because of its easy ignitability and fairly fast burning rate [2], despite of the fact its specific energy release in CO2 is twice less than in O2 combustion, and 3–4 times less than hydrocarbon fuels burning in oxygen. Another problem relevant to flights in Martian atmosphere is that of low density being approximately 60 times less than atmosphere on Earth, which means the Mars surface density is equal to atmospheric density on Earth at the altitude of 15 km. The lift force being directly proportional to atmospheric density provides much less favorable flight conditions. Combining this factor with lower thrust characteristics provided by metal burning in CO2 engines, probably, brought the concept of rover dominating over hopper. The aim of the present paper is reconsidering traditional flight concept for Mars applications.
2. Aerodynamic screen effect Making use of classical airplane flight concept in Martian atmosphere could be hardly realized due to atmospheric density being 60 times less than on Earth. The fact that gravity on Mars surface is only 38% of that on Earth does not compensate the difficulty, because for a typical air-plane weight being decreased 2.6 times the lift force should decrease 60 times for one and the same cruising velocity. The classical aerodynamics concept says that the lift force for subsonic wings is generated due to pressure differential on both sides of wing in streaming flow. The classical formulas for lift force of a single wing do not take into account the presence of other bodies in the flow. However, it is known that wing moving near rigid boundary produces strong effect on the forces acting on wing from streaming fluid [7].
251
At the beginning of the XX century it was observed that the lift force of a wing moving near flat surface increases strongly in comparison with free flight. An article about screen effect by Juriev [8] was published in 1923 in the USSR. That fact was used in creation of new flying devices – screen-planes, which got the Russian name “ekranoplan”. In 1932 Grohovsky constructed a full-scaled model of a new marine flying device – catamaran. At the same time Finnish engineer T. Kaario proceeded to test his flying apparatus that used a screen effect. Then (1963–1976) a Soviet machinery designer R.L. Bartini created a screenplane project SVVP-2500 that took off in 1974. The first Soviet manned jet screen-plane SM-1 was created in collaboration with R. Alekseev in 1960–1961 [9]. Giant screen-plane KM was constructed by 1966 and “Orlyonok” type screen-planes were built from 1974 to 1983.[7] Designing of new flying devices continues in many countries. Sedov obtained an analytical solution for the lift force in terms of Weierstrass functions [10] using theory of a complex variable. Approximate analytical solution of the problem of non-steady plane moving near rigid surface was obtained by Rozjdestvensky [11] using asymptotic expansion. Theoretical investigation of a wing moving near rigid surface was made by Panchenkov [12,13] and Gorelov [14], but the obtained solutions incorporated free constants. Experimental results are shown in [15]. Numericoanalytical solutions were obtained in [16]. Below numerical solution and analytical formulas for the wing flying near rigid surface will be obtained and analyzed for being applied in Martian flight vehicle design.
3. Mathematical problem statement We regard plane wing motion in compressible ideal fluid. The system of equations for fluid flow in the motionless co-ordinates is the Euler system ∂ρ ∂ρ ∂ρ ∂V x ∂V y þV x 0 þ V y 0 þ ρ þ 0 ¼ 0; 0 ∂t ∂x ∂y ∂x ∂y ∂V x ∂V x ∂V x 1 ∂p þV x 0 þV y 0 ¼ ; ∂t ∂x ∂y ρ ∂x0 ∂V y ∂V y ∂V y 1 ∂p þVx 0 þVy 0 ¼ : ∂t ∂x ∂y ρ ∂y0
ð1Þ
k Flow is assumed adiabatic p=p0 ¼ ρ=ρ0 . In case of steady-state flight conditions at a constant altitude h above the surface the system of equations could be regarded in the moving co-ordinate system x ¼ x0 þ Vt; y ¼ y0 (V is the parallel to surface flight velocity), where it takes the stationary form. We assume disturbances being small and attack angle (α) being small as well, thus neglecting the second order of magnitude terms in linearization of system (1), which yields 1 ∂p ∂V x ∂V y þ ρ0 þ 0 ¼ 0; p ¼ p0 þ a2 ρ ρ0 ; 0 2 ∂x ∂y a ∂t ∂V y dp ∂V x 1 ∂p 1 ∂p ¼ ¼ a2 ¼ ; : ð2Þ dρ ρ ¼ ρ 0 ∂t ρ0 ∂x0 ∂t ρ0 ∂y0
252
M. Smirnova / Acta Astronautica 103 (2014) 250–256
For small disturbances system (2) testifies that acceleration vector has potential ∂u ∂ p ∂v ∂ p ¼ 0 ¼ 0 ; ∂t ∂x ρ0 ∂t ∂y ρ0 which yields velocity potential Z t p ∂φ ∂φ φ¼ dt; u ¼ 0 ; v ¼ 0 ∂x ∂y t 0 ρ0
ð3Þ
Boundary conditions for the system of equations include slip conditions on the wing surface and on the ground surface. After transfer to another independent variable pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x~ ¼ x=δ; δ ¼ 1 M 2 the equations and boundary conditions describing the gas flow take the following form:
0 r x~ r L=δ;
p ¼ p0
∂φ ~ ¼ V α 7 ðδxÞ; ∂y ∂φ y ¼ 0; ¼0 ∂y 7
y¼h ;
1 o x~ o 1;
ρ0 V ∂φ : δ ∂x~
ð5Þ
Boundary conditions should be supplemented with function behavior at the infinity for the uniqueness of the solution. Thus an analytical function satisfying boundary conditions and decreasing at the infinity should be developed. 4. Analytical solution and approximate formulas The solution of the Laplace equation can be developed in the form of a real part for the analytical function of ~ yÞ ¼ Re ΦðzÞ; z ¼ x~ þ iy. Actually, a complex variable φðx; it is necessary to develop first derivative of the analytical 0 function, which could be denoted as: iTðzÞ ¼ Φ ðzÞ. The development of the analytical function is reduced to the following boundary problem: 0 r x~ r L=δ;
7
y¼h ;
1 o x~ o 1; y ¼ 0;
~ ¼ V α~ 7 ðxÞ ~ ReTðx~ þ ihÞ ¼ V α 7 ðδxÞ
~ ¼0 ReTðxÞ
The pressure can be developed from the following formula: pðx~ þ iyÞ ¼ p0 þ
0 r x~~ r 1;
~7 y~~ ¼ h~ ;
1o x~~ o1; y~~ ¼ 0;
Thus gas flow with small disturbances in potential field of mass forces is always potential. The solution is equivalent to solution of the system 2 1 ∂2 φ ∂ φ ∂2 φ ∂φ ð4Þ þ 02 ¼ 0; p ¼ p0 ρ0 : 2 2 02 ∂t a ∂t ∂x ∂y
∂2 φ ∂2 φ þ ¼ 0; ∂x~ 2 ∂y2
The development of the analytical function is reduced to the following boundary problem:
ρ0 V ImTðx~ þiyÞ δ
For the uniqueness of the solution boundary conditions should be supplemented with function behavior at the infinity and at the ends of the segment 0 r x~ rL=δ. It is considered that the function tends to zero at the infinity and is limited at the rear edge of the wing, which is the result of the Chaplygin–Zhukovsky hypothesis. The solution is developed with the help of the symmetry principle. The following non-dimensional variables ~ ~ ~ L; are introduced x~~ ¼ x= y~~ ¼ y=L; h~ ¼ h=L; L~ ¼ L=δ.
~~ ~~ ¼ V α~~ 7 ðxÞ ~~ Re Tðx~~ þ ihÞ ¼ V α 7 ðLxÞ
~~ 0 þ Þ ¼ 0 Re T þ ðx;
The tildes will be omitted in the following equations. The solution of the problem can be derived in the form of Cauchy type integral [17]: " # Z 1 Z 1 1 1 μðtÞ 1 ζ ðtÞ þ iηðtÞ dt þ dt TðzÞ ¼ Χ 1 ðzÞ 2π i 0 Χ 1þ ðtÞ t þ ih z 2π i 0 t þ ih z " # Z 1 Z 1 1 1 μðtÞ 1 ζ ðtÞ iηðtÞ dt þ dt Χ 2 ðzÞ þ 2π i 0 Χ 2 ðtÞ t ih z 2π i 0 t ih z where
qffiffiffiffiffiffiffiffiffi z z1 z1 ¼ 1 þ ih; z2 ¼ ih z z2 ; ffiffiffiffiffiffiffiffiffiffiffi ffi s z z1 Χ 2 ðzÞ ¼ ; z1 ¼ 1 ih; z2 ¼ ih z z2
Χ 1 ðzÞ ¼
The function TðzÞ satisfies the boundary condition 1 o x o 1; y ¼ 0; Re T þ ðx; 0 þ Þ ¼ 0. þ It is assumed that μðsÞ ¼ Vðα~~ ðsÞ þ α~~ ðsÞÞ; ζ ðsÞ ¼ þ ~ ~ ~ ~ Vðα ðsÞ α ðsÞÞ. The function TðzÞ will satisfy the 7 boundary conditions 0 rx r 1; y ¼ h ; Re Tðx þ ihÞ ¼ 7 ~ V α~ ðxÞ if the following singular integral equation is fulfilled: Z 1 Z 1 1 ηðtÞ 1 ðt xÞηðtÞ dt dt 2π 0 t x 2π 0 ðt xÞ2 þ 4h2 Z 1 ~þ Vh α~ ðtÞ α~~ ðtÞ dt π 0 ðt xÞ2 þ 4h2 r Z 1 ffiffiffiffiffiffiffiffiffiffi þ V t ðt xÞ ðα~~ ðtÞ þ α~~ ðtÞÞdt RðxÞ sin αðxÞ 2π 0 1 t ðt xÞ2 þ 4h2 Z rffiffiffiffiffiffiffiffiffiffi þ Vh 1 t 1 þRðxÞ cos αðxÞ ðα~~ ðtÞ π 0 1 t ðt xÞ2 þ 4h2 þ α~~ ðtÞÞdt ¼ 0: where 1 2h 2h þ arctg ; αðsÞ ¼ arctg 2 1s s
RðsÞ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 ð1 sÞ þ4h 2
s2 þ4h
:
7 If the wing has a shape of a plate α~~ ¼ γ the singular integral equation takes the following form: Z Z 1 1 ηðtÞ 1 1 ðt sÞηðtÞ dt dt π 0 t s π 0 ðt sÞ2 þ 4h2 Z 1 rffiffiffiffiffiffiffiffiffiffi Vγ t ðt sÞ sin αðsÞRðsÞ dt þ 1 t ðt sÞ2 þ 4h2 π 0 r ffiffiffiffiffiffiffiffiffi ffi Z 1 2Vhγ t dt cos αðsÞRðsÞ ¼0 1 t ðt sÞ2 þ 4h2 π 0
The third and the fourth integrals can be taken analytically using the theory of residues. The following singular integral equation can be determined: Z Z 1 1 η~ ðtÞ 1 1 ðt sÞη~ ðtÞ dt ¼ dt FðsÞ π 0 t s π 0 ðt sÞ2 þ 4h~ 2
M. Smirnova / Acta Astronautica 103 (2014) 250–256
253
where
η~ ðsÞ ¼ ηðsÞ=V γ ;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 ð1 sÞ þ 4h ; FðsÞ ¼ RðsÞ sin αðsÞ 1; RðsÞ ¼ 2 2 s þ 4h 1 2h 2h þ arctg : αðsÞ ¼ arctg 2 1 s s
4.1. Determination of the lift force After the regularization the singular integral equation is reduced to the Fredholm equation: Z 1 η~ ðxÞ þ η~ ðtÞKðx; tÞdt ¼ GðxÞ; 0
Fig. 1. Schematic picture for a model wing profile simulated numerically.
where Kðx; tÞ ¼
1 4π
rffiffiffiffiffiffiffiffiffiffi 1x x
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin αðtÞ δðx; tÞ ; 2 2 RðtÞ ½tð1 xÞ xð1 tÞ þ 4h
rffiffiffiffiffiffiffiffiffiffi Z rffiffiffiffiffiffiffiffiffiffi 1x 1 1 s FðsÞds ; GðxÞ ¼ x π 0 1s sx
δðx; tÞ ¼ arg ½tð1 xÞ xð1 tÞ þ 2hi ; 1 2h 2h ; αðtÞ ¼ arctg þarctg 2 t 1t sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 ð1 tÞ þ 4h RðtÞ ¼ 2 t 2 þ 4h The following formula is used to determine the lift force: Z L Z L=δ
p ðsÞ p þ ðsÞ ds ¼ δ p ðs~ Þ p þ ðs~ Þ ds~ ð6Þ FΠ ¼ 0
0
The pressure is derived from the problem statement: pðs~ Þ ¼ p0 þ ρ0δV Im p ðs~ Þ p þ ðs~ Þ ¼
Tðs~ Þ
ρ0 V ImðT ðs~ Þ T þ ðs~ ÞÞ δ
In the case of a plate α~~ ¼ γ the lift force is determined by formula: " # Z i ρ π LV 2 γ 1 1h 1þ η~ ðs~~ Þ ds~~ ð7Þ FΠ ¼ 0
π
δ
0
If the wing has a model convex shape, for example þ α~~ ðxÞ ¼ 2γ x; α~~ ðxÞ ¼ γ , the lift force derived from (6) has the following form: "Z rffiffiffiffiffiffiffiffiffiffi Z rffiffiffiffiffiffiffiffiffiffiffi # Z 1 ρ0 V 2 γ L 1 1 s 1 1 t 0 2t 0 þ 1 FП ¼ η~ ðsÞds dt ds s π 0 δ 1 t0 t0 s 0 0 "
ρ V 2 γ Lπ 5 1 ¼ 0 4 π δ
Z
1 0
#
η~ ðsÞds
ð8Þ
The derived lift force differs from its analog for boundless surface by an extra summand, which decreases with the increase of the height. Formula (6) provides an estimate for the increase of the lift force on approaching the rigid surface, which is crucial for flights in Martian atmosphere.
5. Numerical solution. For the obtained result being verified the problem of lift force in streaming flow in the vicinity of rigid boundary was solved numerically. Numerical solution of the problem of potential streaming flow of incompressible fluid was performed using collocation method [18], which is based on distributing boundary panels with dipoles and a vortex at the tale end of the contour. The method was modified for solving problems in a bounded space using reflection techniques. The streamed contour ∂Ω is split by N þ1 grid points ξk , kK¼o,…,N so that the first and the last coincide ξo ¼ ξN at the tale end (Fig. 1). A pair of grid points [ξk , ξk þ 1 ] determines the panel segment ∂Ωk, the panel segments approximate entire wing profile. The velocity is given by the following formula: 2 3 N1 uPV ðx; ξk Þ uPV ðx; ξk Þ þ 4 5 uðxÞ ¼ U þ ∑ μk uPV ðx; ξk þ 1 Þ uPV ðx; ξk þ 1 Þ k¼0 h i þ κ uPV ðx; ξ0 Þ uPV ðx; ξ0 Þ is the velocity of undisturbed gas, ξ ¼ ! ! ξ r y ξ¼ ; uPV ¼ 1=2π r 2 is the veloη þr x
where U !
ξ ; η
city of disturbed gas from a vortex with the center in ξ and a unit power, r x ¼ x ξ;
r y ¼ y η;
r 2 ¼ r 2x þ r 2y ;
x ¼ ðx; yÞ
½ξk ; ξk þ 1 ; k ¼ 1:::N are the segments, which approximate the contour. Boundary condition on the rigid surface is satisfied. Unknown coefficients μk ; k can be developed from the streaming conditions on the contour, fulfilled in the centers of the segments. The system of linear algebraic equations is closed with the Chaplygin–Zhukovsky hypothesis, which minimizes the module of the velocity at the rear edge of the contour:
μ0 μN 1 þk ¼ 0
254
M. Smirnova / Acta Astronautica 103 (2014) 250–256
The lift force can be derived from the following equation: I 1 N 1 2 pn dσ ¼ ρ ∑ Δk nk u^ k ; F¼ 2 k¼0 ∂Ω where u^ k ¼ uðx^ k Þ;
x^ j ¼ 12 ðξj þ ξj þ 1 Þ
6. Results and discussion. The dependence of the reduced lift force F Π δ=ρ0 V 2 πγ L of a plate and a convex contour upon the altitude ~ H ¼ h~ ¼ hδ=L, which was developed analytically, is shown in Fig. 2. It can be seen that the lift force decreases with the increase of the height above the rigid surface until the force reaches its magnitude in a boundless medium. The lift force of a convex contour is bigger than that of a plate for the same altitudes. In Fig. 3. the dependence of the reduced lift force ~ F Π =ρ0 V 2 πγ L of a plate upon the altitude H ¼ h~ ¼ hδ=L, which was obtained analytically, is compared with the numerical solution for the Math number M ¼0.2 and for small attack angles less than 31. It is seen that for small attack angles the analytical and numerical solutions coincide, while for large angles surpassing 101 that would not be the case, and numerical solution should be used. Streamlines and the distribution of velocity along them obtained in numerical solution are shown in Fig. 4. for the case of gas flows streaming a plate with an inclination angle α moving at an altitude H ¼ h=L. The velocity module is presented on the scaling map in the right hand side of fig. 4, the measure unit is the velocity of undisturbed gas. It can be seen that the lift force of a plate with an inclination angle less than 301 decreases with the increase of the
Fig. 2. The dependence of the reduced lift force F Π δ=ρ0 V 2 πγL of a plate (lower curve) and a convex contour (upper curve) upon the altitude ~ H ¼ h~ ¼ hδ=L
Fig. 3. The dependence of the reduced lift force F Π =ρ0 V 2 πγL of a plate ~ upon the altitude H ¼ h~ ¼ hδ=L for M ¼ 0.2 obtained analytically (blue curve) and numerically (red curve). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
altitude above the surface. There is no such an effect for inclination angles more than 301. Analysis of the obtained results shows that a motionless surface (screen) essentially affects the lift force at the altitude, which is comparable with the length of a wing. If the altitude above the surface several times surpasses the wing span the screen effect practically disappears and the lift force tends to its value in an unbounded space. The effect of lift force increase on approaching the rigid surface manifests for relatively small attack angles (below 301). For small Mach numbers the lift force coincides with that determined by the classical Zhukovsky solution for incompressible fluid F П ¼ ρ0 V 2 πγ L. The increase of the flight Mach number brings to an increase of the lift force, as well as drag force. The screen effect still manifests in the same proportion. However, the increase of Mach number brings to growing linear theory error ( M2 ), thus on its approaching unity the obtained solution loses validity. Analysis of results presented in Fig. 2 enables to perform the following estimates. On approaching the wing to a plane screen at an altitude comparable with the wing length there appears a small screen effect manifesting in an increase of the lift force 3–4% as compared to that at high altitudes. The further decrease of altitude down to 0:1L brings to an increase of lift force 37–38% as compared to that at high altitudes (34% as compared with the altitude L). If the thickness of a wing is taken into consideration the lift force of a convex wing is bigger than that of a plate. Nevertheless, screen effect manifests for a convex wing as well (Fig. 4), but at a lower level. Estimates show, that for convex wing regarded in the present paper the increase of the lift force on decreasing altitude from L down to 0:1L is only 25%. However, the absolute value of the lift force is
M. Smirnova / Acta Astronautica 103 (2014) 250–256
255
Fig. 4. Streamlines and the distribution of velocity along them in case of gas flows around a plate with an inclination angle α on the altitude H ¼ h=L: (a) α ¼101, H¼ 0,1; (b) α ¼101, H¼ 0,5; (c) α ¼101, H¼ 1; (d) α ¼201, H¼0,5; (e) α ¼201, H¼ 1; (f) α¼ 301, H¼ 0,5; and (g) α¼ 301, H¼ 1;.
higher than that for a plane wing at an altitude 0:1L(surpasses 20%) thus providing the gain of 60% with the reference value. The manifestation of screen effect for convex wings essentially depends on the wing profile. The developed method makes it possible to evaluate screen effect for wings of different profiles. 7. Conceptual design of Mars vehicle Analyzing the Mars flight conditions as compared with that on Earth one can conclude that the conditions are less favorable. Despite of the fact that Mars gravity is 2.65 times less than that on Earth, and sonic speed in Martian atmosphere is 2 times less than that on Earth, which gives a gain in lift capacity for one and the same flight velocity,
the 60 times lower density brings to an overall loss. For example, for flying at a speed of 150 m/s the total loss in lift capacity will be 6 times. However, formulas (7), (8) and results illustrated in Fig. 3 show that the increase of wing span and decrease of flight altitude above the planet surface could compensate that loss and even bring to some gain in lift capacity. This flying vehicle utilizing the screen effect should rather have a discos convex shape. An artistic view of possible screen flight for Martian atmosphere is shown in Fig. 5. The two engines are located on top of the vehicle for avoiding dust sucking while flying near the planet surface. The discos shape provides the higher value for characteristic size parameter L. This shape is considered to be more effective; however the necessity of stabilization and flight control could need additional
256
M. Smirnova / Acta Astronautica 103 (2014) 250–256
First estimate for the shape of perspective vehicle made it possible recommending a convex shape with upper parabolic profile and lower linear profile. The principle is suggested for designing Martian screen-plane vehicles. Acknowledgment. The research was partially supported by Russian Foundation for Basic Research (Grant 12-08-91702). References.
Fig. 5. An artistic view of perspective screen-flight vehicle for Martian atmosphere.
Fig. 6. Screen-plane KM designed by R. Alekseev.
wings and/or tales. It is magnificent, that the cruising flight regime is stable in terms of flight altitude: on decreasing altitude the lift force increases (Fig. 2), and on incidental increase of altitude the lift force decreases thus returning vehicle back to the assigned altitude. The analogous flight vehicles were used on Earth for the purpose of increasing lift capacity in the territory of steppes, deserts and seas [7]. Fig. 6 illustrates one of vehicles used in the region of Caspian Sea. It is seen from the figure that wings have practically square shape, engines are located on top of the front and rear parts, and a big tail for maneuvering is necessary. 8. Conclusions The paper presents an analysis of flight options in Martian atmosphere. It is demonstrated, that using metal combustion rocket engines with carbon dioxide as an oxidizer makes it possible obtaining necessary thrust for flying in Martian atmosphere. Both analytical and numerical solutions determining lift force of a wing moving in the proximity of a rigid screen demonstrate the increase of the lift force on approaching screen, and provide formulas for estimating lift force for a flight vehicle moving in compressible gas near rigid surface.
[1] E. Shafirovich, A. Varma., Metal-CO2 propulsion for Mars missions: current status and opportunities, J. Propuls. Power 24 (3) (2008) 385–394. [2] S. Yuasa, H. Isoda, CO2-Breathing Propulsion for a Mars Airplane, AIAA Paper 89-2863, July 1989. [3] E. Ya. Shafirovich, A.A. Shiryaev, U.I. Goldshleger, Magnesium and carbon dioxide: a rocket propellant for Mars missions, J. Propuls. Power 9 (2) (1993) 197–203. [4] V.F. Nikitin, V.R. Dushin, Y.G. Phylippov, J.C. Legros, Pulse detonation engines: technical approaches, Acta Astronaut. 64 (2009) 281–287. [5] N.N. Smirnov, V.F. Nikitin, S. Alyari Shurekhdeli, Investigation of Selfsustaining waves in metastable systems: deflagration-to-detonation transition, J. Propuls. Power 25 (3) (2009) 593–608. [6] S.V. Khomik, B. Veyssiere, S.P. Medvedev, V. Montassier, G.L. Agafonov, M.V. Silnikov, On some conditions for detonation initiation downstream of a perforated plate, Shock Waves 23 (3) (2013) 207–211. [7] M.N. Smirnova, A.V. Zvyaguin, Theoretical solution for the lift force of ecranoplan moving near rigid surface, Acta Astronaut. 68 (11–12) (2011) 1676–1680. [8] B.N. Juriev, Earth Impact on the Aerodynamical Characteristics of a Wing, «Air fleet herald» №1, 1923. [9] N.I. Belavin, Screen-flights (Ecranoplan), Shipbuilding Publ., Moscow, 1977. [10] L.I. Sedov, Plane Problems of Hydrodynamics and Aerodynamics, Science Publ., Moscow, 1980. [11] K.V. Rozjdestvensky, Asymptotic theory of a wing moving on small distances from a rigid wall, Mech. Fluid Gas (№6) . [12] A.N. Panchenkov, Hydrodynamics of An Underwater Wing, Book Printing House № 5, Kiev, 1964. [13] A.N. Panchenkov, Theory of Optimal Supporting Surface, Science, Novosibirsk, 1983. [14] D.N. Gorelov, Methods of Solving Two-dimensional Boundary Problems in the Theory of a Wing, SO RAN, Novosibirsk, 2000, 215. [15] Y.М. Serebriisky, S.А. Biyachuev, The Study of Horizontal Steady Wing Motion in a Pipe, TSAGI papers (Central Aero Hydrodynamic Institute), issue 437, 1939. [16] K.V. Frolova, V.A. Frolov, Lift force of a plate near screen, in: Samara State Aerospace University Proceedings, 2009. [17] F.D. Gahov, Boundary Problems, Science, Moscow, 1977. [18] C. Pozrikidis, Fluid Dynamics: Theory, Computation and Numerical Simulation, Springer, New York, 2009. Maria Smirnova, born in Moscow in 5 November 1987. Graduated from Moscow M. V. Lomonosov State University in 2010 and got the degree of Master of Science in mechanics of fluids and gas. Submitted thesis in 2013 and got the Ph.D. degree in Physics and Mathematics. Research interests: fluid physics and mathematical physics. She is currently assistant professor in the Faculty of Mechanics and Mathematics, Moscow M.V. Lomonosov State University. At the same time holding half a position of visiting Associate Professor in Saint Petersburg State Polytechnical University. Received Scientific scholarships of O. Deripaska in 2010 and 2012, and scientific scholarship of the President of Russian Federation in 2013.