Turn Performance of an Air-Breathing Hypersonic Vehicle

Turn Performance of an Air-Breathing Hypersonic Vehicle AIAA Aircraft Flight Mechanics Conference Derek J. Dalle, Sean M. Torrez, James F. Driscoll University of Michigan, Ann Arbor, MI 4809 August 8, 0, AFM 0 /6

Why dual-mode ram/scram again? It s that or a rocket! Competition Space access: rocket High-speed, long-range: rocket with glide High-speed loiter:? Maximum practical Mach number is about Specific Impulse [s] 8,000 7,000 6,000 5,000 4,000 3,000,000,000 Turbofan Turbofan with afterburner Ramjet Rocket 0 0 4 6 8 0 Mach Number Theoretical maximum Hydrogen fuel in air Theoretical maximum Hydrocarbon fuel in air For applications where we use lift, doubling the specific impulse is a big increase Wide range of applications Possibility of a large flight envelope, AFM 0 /6

for turning not required Some applications are just a great circle, like high-speed transport. Launch vehicles Either bring all the heavy scramjet equipment into orbit along with the payload (spaceplane) or (turn around and) fly the scramjet part back to the launch area. Military applications Out-and-back trajectories obviously require turning. Even circumnavigation requires some. Then what?, AFM 0 3/6

of engine and vehicle dynamics Energy added Compression Expansion AIR (fast) Air-breathing engine Compressor work AIR + PRODUCTS (faster), AFM 0 4/6

of engine and vehicle dynamics Energy added AIR (fast) compression Air-breathing engine expansion AIR + PRODUCTS (faster), AFM 0 4/6

MASIV/MAS vehicle framework 4 (documented) parameters Takes less than 5 seconds to run Builds vehicle around engine flowpath Two elevators and two rudders >> vehicle = build_vehicle(... 'elevator_span', 0.4,... 'H_combustor_fore', 0.66); >> show_vehicle(vehicle) >> z [m] 0 5 0 5 0 5 30 y [m] 0 5 0 5 x [m], AFM 0 5/6

Canonical engine flowpath 8 a b c d a 3a 4a 4b 5a 6a Several inlet ramps for compression efficiency Arbitrarily shaped, variable-area duct Fuel injection trough any number of ports Jet mixing in combustor with recombination and external expansion, AFM 0 6/6

SAMURI Supersonic Aerodynamic Method Using Riemann Interactions domain initial conditions: p, ρ, T, M physical geometry, AFM 0 7/6

SAMURI Supersonic Aerodynamic Method Using Riemann Interactions Mach wave shock, AFM 0 7/6

SAMURI Supersonic Aerodynamic Method Using Riemann Interactions Mach wave (continuation) 4 3 expansion fan shock (continuation), AFM 0 7/6

SAMURI Supersonic Aerodynamic Method Using Riemann Interactions 4 3 7 5 6 shock slip line shock expansion fan (continuation), AFM 0 7/6

SAMURI Supersonic Aerodynamic Method Using Riemann Interactions, AFM 0 7/6

Example SAMURI solutions A sample inlet solution (temperature in K) y [m] y [m] 0 3..4.6 0 4 6 8 0 4 x [m] 000 900 800 700 600 500 400 300 000 900 800 700 600 500 400 300.5.5 3 3.5 4 x [m], AFM 0 8/6

Example SAMURI solutions Comparison with CFD Solution from CFD++ Solution from SAMURI, AFM 0 9/6

Air Flame Progress Air Mean Contour Flame Progress D equations for temperature, pressure, etc. 3D averaged fuel mixing Fuel Turbulent flame Fuel Averaged turbulent flame Averaged, quasi-steady model Mixing based on jet centerline and spreading Crossflow information collapsed before integration Finite-rate chemistry based on flamelet modeling z Flame Centerline Flame upper boundary Flame lower boundary Lines along which flame properties are computed x, AFM 0 0/6

model Almost the same as the inlet y [m] 0 3 4 8 0 4 6 8 30 x [m] Temperature [K] 500 000 500 000 500, AFM 0 /6

with hypersonic vehicle Three-dimensional vehicle with triangular panels Propulsion flowpath modeled as two-dimensional Vehicle split into propulsive and non-propulsive parts Propulsive panels in red above Vehicle geometry build around engine Neglects 3D phenomena such as lateral spillage Angular velocity not accounted for in engine model, AFM 0 /6

Earth model Equations of motion z e y e x n z n y n Rotating, ellipsoidal Earth default (WGS84) Options for flat or spherical Earth Rotation can be turned on or off independently x e equator prime meridian Submodels Somigliana gravity model 976 standard atmosphere Optional temperature offset Optional wind and wind derivatives, AFM 0 3/6

Flight parameters and control variables State variables, x Position: L, λ, h Attitude: φ, θ, ψ Velocity: M, α, β Angular velocity: p, q, r Can use alternative inputs, such as flightpath angle and heading angle Can specify absolute velocity or wind-relative velocity ẋ = f a (x,u) Control variables, u Equivalence ratio: δ ER Elevators: δ CE, δ DE Rudders: δ CR, δ DR Cowl deflections: δ cx, δ cz, δ ca Fueling location: δ ff Auxilliary parameters, a Wind: w N, w E, w D Temperature offset: T Vehicle design, AFM 0 4/6

Vehicle trim Problem statement: Find x and u such that ẋ = f(x,u) where we pick the value of ẋ beforehand. Usually we want ẋ = 0., AFM 0 5/6

Vehicle trim Problem statement: Find x and u such that ẋ = f(x,u) where we pick the value of ẋ beforehand. Usually we want ẋ = 0. Independent variables and dependent variables We want to pick some of the state variables beforehand, but not all. [ ] T x ξ = T x υ = u Now ξ has independent variables, and υ has dependent variables., AFM 0 5/6

Vehicle trim Problem statement: Find x and u such that ẋ = f(x,u) where we pick the value of ẋ beforehand. Usually we want ẋ = 0. Independent variables and dependent variables We want to pick some of the state variables beforehand, but not all. [ ] T x ξ = T x υ = u Now ξ has independent variables, and υ has dependent variables. Implementation Solving for υ gives us a trim function. υ = g(ξ,ẋ), AFM 0 5/6

Vehicle trim Problem statement: Find x and u such that ẋ = f(x,u) where we pick the value of ẋ beforehand. Usually we want ẋ = 0. Independent variables and dependent variables We want to pick some of the state variables beforehand, but not all. [ ] T x ξ = T x υ = u Now ξ has independent variables, and υ has dependent variables. Implementation Technically that s only an equality constraint. min φ(ξ,υ) subject to υ = g(ξ,ẋ), AFM 0 5/6

High-speed turning flight Slightly altered load factor definition North East Down Normal definition (flat Earth equations of motion), AFM 0 6/6

High-speed turning flight Slightly altered load factor definition North East Down Altered definition (any other equations of motion), AFM 0 6/6

High-speed turning flight Slightly altered load factor definition North East Down Comparison, AFM 0 6/6

System linearization Mathematical concept Find a trimmed condition x = F( x,ū) Calculate the derivative F = F( x + h ie i,ū) F( x,ū) x i h j First-order Taylor series ẋ = F F x + x u u, AFM 0 7/6

System linearization Mathematical concept Find a trimmed condition x = F( x,ū) Calculate the derivative F = F( x + h ie i,ū) F( x,ū) x i h j First-order Taylor series ẋ = F F x + x u u Problem: jagged function Installed thrust coefficient 0.04 0.03 0.0 0.0 0 0.0 0.0 0.03 6 7 8 9 0 Mach number, AFM 0 7/6

System linearization Mathematical concept Find a trimmed condition x = F( x,ū) Calculate the derivative F = F( x + h ie i,ū) F( x,ū) x i h j First-order Taylor series Solution ẋ = F F x + x u u Problem: jagged function Installed thrust coefficient 0.04 0.03 0.0 0.0 0 0.0 0.0 0.03 6 7 8 9 0 Mach number Use a cloud of points around the trimmed condition to create an overdetermined system Use linear least squares to solve for the best-fitting plane Result is hopefully the derivative of the underlying function, AFM 0 7/6

Operating map Non-turning flight conditions h [km] h [km] 3 3 30 9 8 7 6 5 0.4 0.60.6 Equivalence ratio 0.4. 0.6 0.6.4..6.4.8 4 6 7 8 9 0 M 3 3 30 9 8 7 6 Elevator deflection angle [degrees] 0.4 0.60.6 0.4. 0.60.6.4..6.4.8 0.3 0.5 0. 0.5 0. 4 3 h [km] 3 3 30 9 8 7 6 5 0.4 0.60.6 Angle of attack [degrees] 0.4. 0.60.6.4..6.4.8 4 6 7 8 9 0 M 5 4 6 7 8 9 0 M, AFM 0 8/6.5.5 0.5 Calculate trim for a range of altitude and Mach number Example shown for half-fueled MAX- flying east at the equator Dotted lines show dynamic pressure in atmospheres 0

Operating map flight conditions: g-turn h [km] h [km] 0.4 0.60.6 0.4 0.60.6 Equivalence ratio Angle of attack [degrees] 3 3 3 3 5 30 30 4 9 0.6 9 8 8 3 7 0.4 7 6 6 5 0. 5 4 4 6 7 8 9 0 6 7 8 9 0 M M Elevator deflection angle [degrees] Roll angle [degrees] 3 3 3 3 3 65 30.5 30 64.5 9 9 64 8 8 7.5 7 63.5 6 6 63 5 0.5 5 6.5 4 4 6 7 8 9 0 6 7 8 9 0 M M, AFM 0 9/6 0.4. 0.4. 0.60.6.4..6.4.8 0.60.6.4..6.4.8 h [km] h [km] 0.4 0.60.6 0.4 0.60.6 0.4 0.4.. 0.60.6.4..6.4.8 0.6 0.6.4..6.4.8

4 Operating map This time holding dynamic pressure constant at atm.8 Equivalence ratio 0. 0.5 0.3.8 Angle of attack [degrees].5 n [g].6.4. 0.5 6 7 8 9 0 M.8 Elevator deflection angle [degrees] 3.5 3.5.5 0.5 n [g].6.4. 0.5 0 6 7 8 9 0 M.8 Roll angle [degrees] 60 n [g].6.4. 4.5 0 0 0 0 6 7 8 9 0 6 7 8 9 0 M M, AFM 0 0/6 n [g].6.4. 50 40 30 0 0

Pole/zero plots Mach 8 flight, L = 0, h = 6km, σ = 90, n = Longitudinal dynamics Lateral-directional dynamics 0.05 6 4 Imaginary part 0 Imaginary part 0 4 0.05 5 0 5 0 5 0 5 Real part 6 4 0 4 Real part Above example for M = 8, h = 6km, no turning Can be calculated for any flight condition Consistent with expected behavior, AFM 0 /6

Pole/zero plots Mach 8 flight, L = 0, h = 6km, σ = 90, n = Imaginary part 0. 0.05 0 0.05 Longitudinal dynamics Imaginary part 0.5 0 0.5 Lateral-directional dynamics 0. 5 0 5 0 5 0 5 Real part 5 0 5 Real part Above example for M = 8, h = 6km, g-turn Can be calculated for any flight condition Differs from non-turning condition, AFM 0 /6

Primary The vehicle can turn. We can generate operating maps., AFM 0 3/6

Primary The vehicle can turn. We can generate operating maps. Other Development of a low-order vehicle model Stability substantially affected by amount of turning Effects of turning on operating map Verification of inlet design ideas, AFM 0 3/6

Acknowledgments This research was supported by U.S. Air Force Research Laboratory grant FA 8650-07--3744 for the Michigan Air Force Research Laboratory Collaborative Center for Control Science. Mike Bolender as technical monitor Scott G. V. Frendreis This research was completed as part of the Michigan/AFRL Collaborative Center for Control Science, AFM 0 4/6

Mode shapes and properties Mach 8 flight, L = 0, h = 6km, σ = 90, n = State variables: h, altitude; V, total velocity; α, angle of attack; β, sideslip angle; φ, roll angle; θ, pitch angle; ψ, yaw angle; p, q, r, angular velocity Short period mode Pole Time to double Variables.46 0.3 α, q.07 0.38 α, q Spiral mode Dutch-roll mode Pole Time to double Variables 3.87 β, r 3.83 β, r Roll mode Pole Time to double Variables 0.0036 9 φ Phugoid-altitude mode Pole Time to double Variables.5 p Pole Time to double Damping ratio Variables.8 0 3 45.6 h, V.49 0 4 ± 4.5 0 j 5.49 0 h, θ, AFM 0 5/6

Mode shapes and properties Mach 8 flight, L = 0, h = 6km, σ = 90, n = State variables: h, altitude; V, total velocity; α, angle of attack; β, sideslip angle; φ, roll angle; θ, pitch angle; ψ, yaw angle; p, q, r, angular velocity Short period mode Pole Damping Variables 0.35 ± 0.090j 0.97 α, q Spiral mode Pole Time to double Variables 0.36.93 φ Phugoid-altitude mode Dutch-roll mode Pole Time to double Variables 4. 0.7 β, r 4.5 0.7 β, r Roll mode Pole Time to double Variables.5 p Pole Time to double Damping ratio Variables.6 0 3 440 h, V.99 0 ± 3.5 0 j 0.49 h, θ, AFM 0 6/6