MAE 5360: Hypersonic Airbreathing Engines Simplified Internal and External Flow Modeling Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

Dynamic Pressure for Compressible Flows Dynamic pressure, q = ½rV2 For high speed flows, where Mach number is used frequently, convenient to express q in terms of pressure p and Mach number, M, rather than r and V Derive an equation for q = q(p,M)

Summary of Total Conditions If M > 0.3, flow is compressible (density changes are important) Need to introduce energy equation and isentropic relations Must be isentropic Requires adiabatic, but does not have to be isentropic

Review: Normal Shock Waves Upstream: 1 M1 > 1 V1 p1 r1 T1 s1 p0,1 h0,1 T0,1 Downstream: 2 M2 < 1 V2 < V1 P2 > p1 r2 > r1 T2 > T1 s2 > s1 p0,2 < p0,1 h0,2 = h0,1 T0,2 = T0,1 (if calorically perfect, h0=cpT0) Typical shock wave thickness 1/1,000 mm

Summary of Normal Shock Relations Normal shock is adiabatic but nonisentropic Equations are functions of M1, only Mach number behind a normal shock wave is always subsonic (M2 < 1) Density, static pressure, and temperature increase across a normal shock wave Velocity and total pressure decrease across a normal shock wave Total temperature is constant across a stationary normal shock wave

Tabulation of Normal Shock Properties

Summary of Normal Shock Properties

Normal Shock Total Pressure Loss Example: Supersonic Propulsion System Engine thrust increases with higher incoming total pressure which enables higher pressure increase across compressor Modern compressors desire entrance Mach numbers of around 0.5 to 0.8, so flow must be decelerated from supersonic flight speed Process is accomplished much more efficiently (less total pressure loss) by using series of multiple oblique shocks, rather than a single normal shock wave As M1 ↑ p02/p01 ↓ very rapidly Total pressure is indicator of how much useful work can be done by a flow Higher p0 → more useful work extracted from flow Loss of total pressure are measure of efficiency of flow process

Attached vs. Detached Shock Waves

Normal shock wave model still works well Detached Shock Wave Normal shock wave model still works well

Examples of Schlieren Photographs

Oblique Shock Wave Analysis Upstream: 1 M1 > 1 V1 p1 r1 T1 s1 p0,1 h0,1 T0,1 Downstream: 2 M2 < M1 (M2 > 1 or M2 < 1) V2 < V1 P2 > p1 r2 > r1 T2 > T1 s2 > s1 p0,2 < p0,1 h0,2 = h0,1 T0,2 = T0,1 (if calorically perfect, h0=cpT0) q b

Oblique Shock Control Volume Analysis Split velocity and Mach into tangential (w and Mt) and normal components (u and Mn) V·dS = 0 for surfaces b, c, e and f Faces b, c, e and f aligned with streamline (pdS)tangential = 0 for surfaces a and d pdS on faces b and f equal and opposite Tangential component of flow velocity is constant across an oblique shock (w1 = w2)

Summary of Shock Relations Normal Shocks Oblique Shocks

q-b-M Relationship Shock Wave Angle, b Detached, Curved Shock Strong M2 < 1 Weak Shock Wave Angle, b M2 > 1 Detached, Curved Shock Deflection Angle, q

Some Key Points For any given upstream M1, there is a maximum deflection angle qmax If q > qmax, then no solution exists for a straight oblique shock, and a curved detached shock wave is formed ahead of body Value of qmax increases with increasing M1 At higher Mach numbers, straight oblique shock solution can exist at higher deflection angles (as M1 → ∞, qmax → 45.5 for g = 1.4) For any given q less than qmax, there are two straight oblique shock solutions for a given upstream M1 Smaller value of b is called the weak shock solution For most cases downstream Mach number M2 > 1 Very near qmax, downstream Mach number M2 < 1 Larger value of b is called the strong shock solution Downstream Mach number is always subsonic M2 < 1 In nature usually weak solution prevails and downstream Mach number > 1 If q =0, b equals either 90° or m

Examples Incoming flow is supersonic, M1 > 1 If q is less than qmax, a straight oblique shock wave forms If q is greater than qmax, no solution exists and a detached, curved shock wave forms Now keep q fixed at 20° M1=2.0, b=53.3° M1=5, b=29.9° Although shock is at lower wave angle, it is stronger shock than one on left. Although b is smaller, which decreases Mn,1, upstream Mach number M1 is larger, which increases Mn,1 by an amount which more than compensates for decreased b Keep M1=constant, and increase deflection angle, q M1=2.0, q=10°, b=39.2° M1=2.0, q=20°, b=53° Shock on right is stronger

Oblique Shocks and Expansions Prandtl-Meyer function, tabulated for g=1.4 in Appendix C (any compressible flow text book) Highly useful in supersonic airfoil calculations

Prandtl-Meyer Function and Mach Angle

Swept Wings in Supersonic Flight If leading edge of swept wing is outside Mach cone, component of Mach number normal to leading edge is supersonic → Large Wave Drag If leading edge of swept wing is inside Mach cone, component of Mach number normal to leading edge is subsonic → Reduced Wave Drag For supersonic flight, swept wings reduce wave drag

Wing Sweep Comparison F-100D English Lightning

Swept Wings Example M∞ < 1 SU-27 q M∞ > 1 ~ 26º m(M=1.2) ~ 56º

Supersonic Inlets Normal Shock Diffuser Oblique Shock Diffuser

EFFECT OF MASS FLOW ON THRUST VARIATION Mass flow into compressor = mass flow entering engine Re-write to eliminate density and velocity Connect to stagnation conditions at station 2 Connect to ambient conditions Resulting expression for thrust Shows dependence on atmospheric pressure and cross-sectional area at compressor or fan entrance Valid for any gas turbine

NON-DIMENSIONAL THRUST FOR A2 AND P0 Thrust at fixed altitude is nearly constant up to Mach 1 Thrust then increases rapidly, need A2 to get smaller

Supersonic and Hypersonic Vehicles

SUPERSONIC INLETS At supersonic cruise, large pressure and temperature rise within inlet Compressor (and burner) still requires subsonic conditions For best hthermal, desire as reversible (isentropic) inlet as possible Some losses are inevitable

REPRESENTATIVE VALUES OF INLET/DIFFUSER STAGNATION PRESSURE RECOVERY AS A FUNCTION OF FLIGHT MACH NUMBER

C-D NOZZLE IN REVERSE OPERATION (AS A DIFFUSER) Not a practical approach!

C-D NOZZLE IN REVERSE OPERATION (AS A DIFFUSER)

Example of Supersonic Airfoils http://odin.prohosting.com/~evgenik1/wing.htm

Supersonic Airfoil Models Supersonic airfoil modeled as a flat plate Combination of oblique shock waves and expansion fans acting at leading and trailing edges R’=(p3-p2)c L’=(p3-p2)c(cosa) D’=(p3-p2)c(sina) Supersonic airfoil modeled as double diamond Combination of oblique shock waves and expansion fans acting at leading and trailing edge, and at turning corner D’=(p2-p3)t

Approximate Relationships for Lift and Drag Coefficients

http://www. hasdeu. bz. edu http://www.hasdeu.bz.edu.ro/softuri/fizica/mariana/Mecanica/Supersonic/home.htm CASE 1: a=0° Shock waves Expansion

CASE 1: a=0°

CASE 2: a=4° Aerodynamic Force Vector Note large L/D=5.57 at a=4°

CASE 3: a=8°

CASE 5: a=20° At around a=30°, a detached shock begins to form before bottom leading edge

CASE 6: a=30°

Example Question Compare with your solution Consider a diamond-wedge airfoil as shown below, with half angle q=10° Airfoil is at an angle of attack a=15° in a Mach 3 flow. Calculate the lift and wave-drag coefficients for the airfoil. Compare with your solution

Compressible Flow Over Airfoils: Linearized Flow, Subsonic Case

Review True for all flows: Steady or Unsteady, Viscous or Inviscid, Rotational or Irrotational Continuity Equation 2-D Incompressible Flows (Steady, Inviscid and Irrotational) 2-D Compressible Flows (Steady, Inviscid and Irrotational) steady irrotational Laplace’s Equation (linear equation) Does a similar expression exist for compressible flows? Yes, but it is non-linear

STEP 1: VELOCITY POTENTIAL → CONTINUITY Flow is irrotational x-component y-component Continuity for 2-D compressible flow Substitute velocity into continuity equation Grouping like terms Expressions for dr?

STEP 2: MOMENTUM + ENERGY Euler’s (Momentum) Equation Substitute velocity potential Flow is isentropic: Change in pressure, dp, is related to change in density, dr, via a2 Substitute into momentum equation Changes in x-direction Changes in y-direction

RESULT Velocity Potential Equation: Nonlinear Equation Compressible, Steady, Inviscid and Irrotational Flows Note: This is one equation, with one unknown, f a0 (as well as T0, P0, r0, h0) are known constants of the flow Velocity Potential Equation: Linear Equation Incompressible, Steady, Inviscid and Irrotational Flows

HOW DO WE USE THIS RESULTS? Velocity potential equation is single PDE equation with one unknown, f Equation represents a combination of: Continuity Equation Momentum Equation Energy Equation May be solved to obtain f for fluid flow field around any two-dimensional shape, subject to boundary conditions at: Infinity Along surface of body (flow tangency) Solution procedure (a0, T0, P0, r0, h0 are known quantities) Obtain f Calculate u and v Calculate a Calculate M Calculate T, p, and r from isentropic relations

WHAT DOES THIS MEAN, WHAT DO WE DO NOW? Linearity: PDE’s are either linear or nonlinear Linear PDE’s: The dependent variable, f, and all its derivatives appear in a linear fashion, for example they are not multiplied together or squared No general analytical solution of compressible flow velocity potential is known Resort to finite-difference numerical techniques Can we explore this equation for a special set of circumstances where it may simplify to a linear behavior (easy to solve)? Slender bodies Small angles of attack Both are relevant for many airfoil applications and provide qualitative and quantitative physical insight into subsonic, compressible flow behavior Next steps: Introduce perturbation theory (finite and small) Linearize PDE subject to (1) and (2) and solve for f, u, v, etc.

HOW TO LINEARIZE: PERTURBATIONS

INTRODUCE PERTURBATION VELOCITIES Perturbation velocity potential: same equation, still nonlinear Re-write equation in terms of perturbation velocities: Substitution from energy equation: Combine these results…

RESULT Linear Non-Linear Equation is still exact for irrotational, isentropic flow Perturbations may be large or small in this representation

HOW TO LINEARIZE Limit considerations to small perturbations: Slender body Small angle of attack

HOW TO LINEARIZE Compare terms (coefficients of like derivatives) across equal sign Compare C and A: If 0 ≤ M∞ ≤ 0.8 or M∞ ≥ 1.2 C << A Neglect C Compare D and B: If M∞ ≤ 5 D << B Neglect D Examine E E ~ 0 Neglect E Note that if M∞ > 5 (or so) terms C, D and E may be large even if perturbations are small B A C D E

RESULT After order of magnitude analysis, we have following results May also be written in terms of perturbation velocity potential Equation is a linear PDE and is rather easy to solve Recall: Equation is no longer exact Valid for small perturbations: Slender bodies Small angles of attack Subsonic and Supersonic Mach numbers Keeping in mind these assumptions equation is good approximation

BOUNDARY CONDITIONS Solution must satisfy same boundary conditions Perturbations go to zero at infinity Flow tangency

IMPLICATION: PRESSURE COEFFICIENT, CP Definition of pressure coefficient CP in terms of Mach number (more useful compressible form) Introduce energy equation (§7.5) and isentropic relations (§7.2.5) Write V in terms of perturbation velocities Substitute into expression for p/p∞ and insert into definition of CP Linearize equation Linearized form of pressure coefficient, valid for small perturbations

HOW DO WE SOLVE EQUATION Note behavior of sign of leading term for subsonic and supersonic flows Equation is almost Laplace’s equation, if we could get rid of b coefficient Strategy Coordinate transformation Transform into new space governed by x and h In transformed space, new velocity potential may be written

TRANSFORMED VARIABLES (1/2) Definition of new variables (determining a useful transformation is done by trail and error, experience) Perform chain rule to express in terms of transformed variables

TRANSFORMED VARIABLES (2/2) Differentiate with respect to x a second time Differentiate with respect to y a second time Substitute in results and arrive at a Laplace equation for transformed variables Recall that Laplace’s equation governs behavior of incompressible flows Shape of airfoil is same in transformed space as in physical space Transformation relates compressible flow over an airfoil in (x, y) space to incompressible flow in (x, h) space over same airfoil

FINAL RESULTS Insert transformation results into linearized CP Prandtl-Glauert rule: If we know the incompressible pressure distribution over an airfoil, the compressible pressure distribution over the same airfoil may be obtained Lift and moment coefficients are integrals of pressure distribution (inviscid flows only) Perturbation velocity potential for incompressible flow in transformed space

OBTAINING LIFT COEFFICIENT FROM CP

IMPROVED COMPRESSIBILITY CORRECTIONS Prandtl-Glauret Shortest expression Tends to under-predict experimental results Account for some of nonlinear aspects of flow field Two other formulas which show excellent agreement Karman-Tsien Most widely used Laitone Most recent

Compressible Flow Over Airfoils: Linearized Flow, Supersonic Case

SMALL PERTURBATION VELOCITY POTENTIAL EQUATION Equation is a linear PDE and easy to solve Recall: Equation is no longer exact Valid for small perturbations Slender bodies Small angles of attack Subsonic and Supersonic Mach numbers Keeping in mind these assumptions equation is good approximation Nature of PDE: Subsonic: (1 - M∞2) > 0 (elliptic) Supersonic: (1 - M∞2) < 0 (hyperbolic)

SUPERSONIC APPLICATION Linearized small perturbation equation Re-write for supersonic flow Solution has functional relation May be any function of (x - ly) Perturbation potential is constant along lines of x – ly = constant

DERIVATION OF PRESSURE COEFFICIENT, CP Solutions to hyperbolic wave equation Velocity perturbations Eliminate f’ Linearized flow tangency condition at surface Linearized definition of pressure coefficient Combined result Positive q: measured above horizontal Negative q: measured below horizontal

KEY RESULTS: SUPERSONIC FLOWS Linearized supersonic pressure coefficient Expression for lift coefficient Thin airfoil or arbitrary shape at small angles of attack Expression for drag coefficient