Notes on Classical Sonic Boom Theory Harvey S. H. Lam Professor Emeritus Department of Mechanical and Aerospace Engineering Princeton University, Princeton, NJ 08544 U. S. A. email: [email protected] http://www.princeton.edu/∼lam March 22, 2001 Abstract A supersonic aircraft generates a wave system which propagates away from the aircraft. This wave disturbance in the far-field is called the sonic boom. The classical sonic boom theory is capable of predicting the strength of the sonic boom in the far-field of a uniform atmosphere as a function of the weight and the length of the aircraft, and the geometry of the wings and fuselage. Recently, the question is asked: if one is allowed to inject disturbances ahead of the aircraft (by energy addition using lasers or electron beams), could one substantially reduce the strength of the sonic boom in the far-field? These notes briefly summarizes the classical sonic boom theory in a uniform atmosphere.1 It is hoped that it can be helpful to those who are assessing the question of current interest. The latest version can be found on the internet at http://www.princeton.edu/∼mae/SHL/SonicBoom.pdf 1A

version dated earlier than March 20, 2001 was distributed to a limited number of people. It contains a minor error: the expression for κ(Mo ) was incorrect. If you have a copy of the earlier version, please destroy it.

1

1

Introduction

The continuity equation is: 1 Dρ +∇·q=0 ρ Dt

(1)

where q is the fluid velocity vector. Assuming thermodynamic equilibrium, we have: Dp Dρ = a2 + Dt Dt

µ

∂p ∂s

¶

ρ

Ds Dρ = a2 + O(∆s) Dt Dt

(2)

where a is the isentropic speed of sound: a2 = (∂p/∂ρ)s = γRT = γp/ρ.

(3)

The O(∆s) term represents the effects of local increase of fluid entropy (along a streamline). For example, localized heat injection ahead of the aircraft, using electron beams or lasers, could be represented by delta functions. Using (2) in (1), with the help of (3), we have: 1 Dp + ∇ · q = O(∆s). ρa2 Dt

(4)

The momentum equation is: Dq 1 = − ∇p + ν∇2 q Dt ρ

(5)

where ν is the kinematic viscosity. We are temporarily keeping the viscous term (even though it is negligible outside of shock waves) in anticipation of later developments. We shall assume that the incoming freestream has uniform entropy, and that the disturbance are “small.” Under conditions of practical interest, the flow is expected to be homentropic and irrotational. We now introduce the perturbation velocity potential φ: q = Uo (ex + ∇φ).

(6)

1 Dp + ∇2 φ = O(∆s) ρa2 Dt

(7)

Equation (4) becomes:

2

For steady flow, the substantial derivative is simply: D = q · ∇. Dt

(8)

We can compute the substantial derivative of p using the momentum equation as follows: Dp Dt

=

ρq · ∇p0 ρ 

(9) −q·∇q

z

}|



{

  q·q 2  = ρq ·  q × (∇ × q) − ∇( 2 ) +ν∇ q ³

´

= ρq · −q · ∇q + ν∇2 q

(10) (11)

We now use cylindrical coordinates (x, r, θ), and denote the components of the velocity vector by (u, v 0 , $ 0 ). We shall assume that in the far-field the θ dependence and $ are both negligible. We have: 1 Dp ∂u ∂v 0 ∂u ≈ −u(u + v0 ) − v0u + νu∇2 u + O(...) ρ Dt ∂x ∂x ∂r

(12)

where O(...) represents all “higher order” terms, including those caused by local entropy production ∆s. The leading order approximation (keeping only linear terms) to (12) is: 1 Dp ∂ 2φ ≈ −Uo2 2 + O(...) (13) ρ Dt ∂x Using this in (7), we have the linear governing equation for φ: (1 −

∂ 2φ Mo2 ) 2 ∂x

µ

1 ∂ ∂φ + r r ∂r ∂r

¶

+ λ∇

2

µ

∂φ ∂x

¶

= O(...)

(14)

where λ, which has the dimension of length (of the order of a few mean free paths in units of L), is defined by λ≡

νUo , a2o

(15)

and Mo is the flight Mach number: Mo ≡

3

Uo . ao

(16)

The next approximation to (12) (neglecting third order and higher terms) is: 1 Dp ∂2φ ∂φ ∂ 2 φ ≈ −u2 2 − 2Uo2 + νUo ∇2 ρ Dt ∂x ∂r ∂x∂r

µ

∂φ ∂x

¶

+ O(...)

(17)

Equation (14) can now be written as: (M 2 − 1)

µ

∂2φ 1 ∂ ∂φ − r ∂x2 r ∂r ∂r

¶

+ 2Mo2

µ

∂φ ∂ 2 φ ∂φ = λ∇2 ∂r ∂x∂r ∂x

¶

+ O(...).

(18) This is now a nonlinear PDE. In addition to the new last term on the left hand side, the other nonlinear term comes from M (in the paranthesis of the first term); it is the local flow Mach number, u/a, and not Mo . The small difference between these two Mach numbers is crucial to the far-field theory.

2

Characteristic Coordinates

We now introduce new independent variables (ξ, η): q

ξ = x − r Mo2 − 1,

(19)

η = x + r Mo2 − 1,

(20)

q

and consider the flow field in the neighborhood of the Mach cone in the far-field: ξ = O(1), η >> 1. (21) We further assume that the dependence of φ on η for fixed ξ is weak, so that taking partial derivative with respect to η lowers the order of magnitude of the entity. We thus have: ∂ ∂x ∂2 ∂x2 ∂ ∂r

z}|{

=

∂ ∂ + ∂ξ ∂η

=

∂2 ∂2 ∂2 + 2 + ∂ξ 2 ∂ξ∂η ∂η 2

(23)

=

q

(24)

(22)



z}|{

z}|{

∂ ∂  Mo2 − 1 − + ∂ξ ∂η

4

∂2 ∂r2 ∂2 ∂x∂r

 

= (Mo2 − 1)  =

q

∂2 ∂ξ 2



−2

∂2 ∂ξ∂η

z}|{ ∂2  + 2

(25)

∂η

z}|{ ∂2   Mo2 − 1 − 2 + 2  ∂ξ ∂η

∂2

(26)

Neglecting terms with overbrace in the far field, we can show that the leading order is:2 ∂P P ∂P ∂ 2P 1 + − κP = ² 2 + O(..., 2 ) ∂η 2η ∂ξ ∂ξ η

(27)

where P

∂φ p0 = , ∂x ρo Uo2 Mo4 (γ + 1) = κ(Mo ), 4(Mo2 − 1)

= −

κ =

(28) ²=

λ 4(Mo2 − 1)

.

(29)

Now we further introduce new dependent and independent variables: p Φ(X, τ ; θ) X = ξ, τ = 2η, P = . (30) κτ The final leading order far-field equation for Φ is: ∂Φ ∂Φ ∂2Φ −Φ = ²τ + (...) ∂τ ∂X ∂X 2

(31)

which is the well known Burgers’ equation (which usually omits the τ dependence on the right hand side). To simplify the notations, we shall apply the initial condition at τ = 0 (instead of τ = O(1)): Φ(X, 0; θ) = f (X). 2 Physically,

(32)

the “new” approximation being made here is: only outgoing waves propagating along ξ=cosntat are being considered. All terms responsible for waves propagating along η=constant are neglected. The original derivation was given by G. B. Whitham in Proc. Roy. Soc. A201, 89, (1950). The derivation of κ(Mo ) requires much algebra, particularly in getting the relation M 2 = Mo2 + Mo2 (2 + (γ − 1)Mo2 )∂φ/∂x via the energy equation. When Sir Lighthill presented the derivation in Volume 6 of the Princeton High Speed Series, General Theory of High Speed Aerodynamics edited by W. R. Sears, he said on page 436: “(Whitham) showed that the equation of continuity can be approximated with some appearance of reason ...”.

5

where f (X) for (31) is to be derived from classical linear wing theory valid in the near field3 , and it is expected to depend on Mo , the geometry and the appropriately normalized weight of the aircraft and the azimuthal angle θ. The needed “boundary conditions” are the vanishing of Φ for X < 0 upstream of the Mach cone from the forward “tip,” and X > 1, (sufficiently) far downstream of the Mach cone from the “tail” of the aircraft. As it now stands, the far-field solution Φ(X, τ >> 1; θ) is governed by (31) and the only input information needed is f (X). The question is: Can one introduce some disturbances in front of f (X) (caused by ∆s due to energy addition) so that the far-field solution Φ(X, τ >> 1; θ) is substantially weaker than having the original f (X) alone?

3

Properties of Burgers’ Equation

Burgers’ equation has some well-known properties.

3.1

Exact solutions

The following are exact solutions which can easily be verified: Φ = C = constant X∗ − X Φ = , X∗ , τ∗ = constants. τ − τ∗

3.2

(33) (34)

Exact Inviscid Solutions

The coefficient ²τ of the second order term in the Burgers’ equation is, from the practical point of view, always a small number even when we consider τ to be “large.” When the right hand side of (31) is neglected, we have the iniviscid Burgers’ equation which can be solved exactly. The result is difficult to express in equations, but it is simple to state in words: A solution point (Φ, X) on the Φ, X plane moves with constant horizontal velocity −Φ. 3

See G. N. Ward’s book on Linearized Wing Theory.

6

Thus, positive solution points moves to the left, while negative solution points move to the right. The exact solution as given by (34) shows the straight solution line rotates (about X∗ with increasing time) counterclockwise when τ > τ∗ , and clockwise when τ < τ∗ . Note that when τ∗ is positive (the slope of the solution line is positive), the solution becomes singular (i.e. a shock forms) at τ = τ∗ .

3.3

Conservation Laws and Shocks

Integrating (31) with respect to X while holding τ fixed between X = A and X = B, we have: d dt

Z B A

"

Φ2 ∂Φ Φ(X, τ ; θ)dX = + ²τ 2 ∂X

#X=B

.

(35)

X=A

• Since the boundary condition for Φ requires that Φ vanishes at upstream and downstream infinity, we have: Z −∞ ∞

Φ(X, τ ; θ)dX = I∞ = constant

(36)

• If the viscous term is neglected, i.e. in the ²τ → 0 limit, the area under the solution curve between two zero crossing points (A and B) is conserved: Z B A

Φ(X, τ ; θ)dX = I(A,B) = constant

(37)

provided Φ(X = A) = Φ(X = B) = 0, even if one or more shocks (see next bullet) are present between A and B. • If we change to a moving frame (Z, τ ) with constant velocty Us in the −X direction, Z = X + Usτ,

τ = τ.

(38)

(31) becomes: ∂Φ ∂Φ ∂ 2Φ − (Φ − Us ) = ²τ + (...). (39) ∂τ ∂Z ∂Z 2 Integrating this equation with respect to Z while holding τ constant as before, we have: d dt

Z B A

"

(Φ − Us )2 ∂Φ Φ(Z, τ ; θ)dZ = + ²τ 2 ∂X

7

#Z=B Z=A

.

(40)

Now, let A and B straddles a discontinuity (i.e. a shock). The left hand side vanishes in the limit as B → A; hence the right hand side must also vanish. Neglecting the viscous term, we obtain the “jump condition” for a finite strength shock wave: Us =

ΦA + ΦB . 2

(41)

In other words, a discontinuity moves with velocity equal to the average values of Φ straddling the discontinuity. If Us is positive, it moves to the left, otherwise it moves to the right. • Given a continuous initial condition f (X) which is positive (without loss of generality) between two zero crossing points (XL , XR ). An incipient shock will emerge when τ reaches the reciprocal of the maximum positive slope of f (X) inbetween. Marking the points which will become the front and the rear of an emerged shock at some later time τ = τ∗ by (XA , ΦA ) and (XB , ΦB ) on a f (X) vs. X plot, it can be shown that XA , XB and τ∗ are related by: ¶ Z XB µ X − X∗ f (X) − dX = 0 (42) τ∗ XA where

ΦB − ΦA >0 XB − XA ΦA (Xb − XA ) = XA − , ΦB − ΦA

τ∗ = X∗

(43) (44)

and ΦA = f (XA ), ΦB = f (XB ). Geometrically, (42) says that the net area between f (X) and a straight line (with positive slope) joining the front and rear points of a shock (when τ = τ∗ > 0) is zero.

3.4

The N-Wave

Consider the initial condition: 1 f (X; θ) = σ( − X), |X| ≤ 1, 2 f (X; θ) = 0, |X| > 1,

σ > 0,

(45) (46)

where σ is some function of Mo , weight and geometry of the aircraft and the azimuthal angle θ. In other words, the initial condition looks

8

like the capital letter N: there is a shock at X = −1, followed by an expansion fan (represented by a straight line with negative slope), and then followed by another (and relatively weaker) shock at X = +1. Comparing this initial condition with the known exact solution (34), we have X∗ = 0.5 and τ∗ = −1/σ < 0. Simply calculation yields the following conserved quantities: I∞ = σ, 9σ I(−1,1/2) = , 8 σ I(1/2,1) = − 8

(47) (48) (49)

The solution is given by a straight line which rotates counterclockwise,and is straddled by two shock waves which moves away from the center. The position of the shock wave is easily determined by honoring the conservation laws: Φ(X, τ ; θ) =

1 2

−X , τ + 1/σ

XF (τ ) ≤ X ≤ XR (τ ),

(50)

where XF (τ ) and XR (τ ) are the positions of the front and rear shocks (determined by honoring the conservation laws): √ 1 − 3 1 + στ XF (τ ) = , (51) √2 1 + 1 + στ XR (τ ) = . (52) 2 The front and rear shock strengths are thus given by: ΦF ΦR

3σ = Φ(XF , τ ; θ) = √ , 2 1 + στ σ = Φ(XR , τ ; θ) = − √ . 2 1 + στ

(53) (54)

Note that both ΦF and ΦR decay with increasing τ . The total distance between the front and rear shock is: √ XR − XF = 2 1 + στ . (55) The evolution of the N-wave can thus be described as follows:

9

The null point separating the front compressive lobe from the rear expansive lobe, X = 0.5, remains fixed in time. The linear part of the solution rotates counterclockwise, its slope decreasing with time. The shocks move away from the null point parabolically. The strength of the front shock, as represented by PF , has the following τ dependence: PF =

Φ(XF (τ ), τ ; θ) 3σ 1 √ = κτ 2κ τ 1 + στ

Now, τ is related to r by: τ=

p

2η = =

r r

q

(56)

2(x + r Mo2 − 1)

(57)

2(ξ + 2r Mo2 − 1)

(58)

q

≈ 2(Mo2 − 1)1/4 r1/2

(59)

So the r dependence of ΦF is given approximately by:4 3σ 1 q PF = (60) 4κ (M 2 − 1)1/4r1/2 1 + 2σ(M 2 − 1)1/4 r1/2 o

o

Thus, the decay of ΦF with respect to r starts with r−1/2 in the intermediate far-field as given by the linear theory, and gradually shifts over to the r−3/4 power for asymptotically large r for the far far-field via nonlinear effects. The quantitative behavior of the rear shock is entirely similar. It is interesting to note that if some disturbance were introduced in front of the compressive lobe, the far-field solution for the compressive lobe is not affected until its front shock catches up with the injected disturbance, while the far field solution for the expansive lobe is totally unaffected.

3.5

What Happens When ²τ = O(1)?

We then have the full (viscous) Burgers Equation. The discontinuities are now smooth transitions (with thickness O(²τ )). While I∞ remains conserved, the area under two zero-crossing points of f (X) will no longer be conserved—it will decay. 4 Note

that both σ and κ depends on Mo . The dependence of the amplitude of ΦF on Mo is not explicitly shown here.

10

4

Wave Drag and f (X; θ)

The wave drag Dw of a supersonic aircraft can be computed from the perturbation x-momentum flux out of a large cylindrical control volume centered at the aircraft: Dwave = − lim

Z +∞ Z 2π

r→∞ −∞

0

ρo u0 v 0 rdθdx

(61)

where v 0 here represents radial perturbation velocity. Using the linear intermediate field solution, we have: Dwave ∝ ρUo2 A

Z +∞ Z 2π −∞

f 2 (X; θ)dθdX

(62)

0

where A has the dimension of area (when f and X are dimensionless). The important point is: Dw is always positive, and any increase of the original f (X; θ) will always increase the wave drag. In additional to the wave drag, there is also a perturbation xmomentum flux out of the rear control surface of this big cylindrical control volume. But this “wake” drag, which includes the vorticity (induced) drag due to lift, contributes nothing to the sonic boom.

5

Discussion

In practical application of this, one must remove the uniform atmosphere simplification. This generalization introduces much mathematical difficulty, for the effects of defraction involves ray tracing which is essentially numerical. For the issues being addressed here, it is reasonable to assume that any effort that wish to claim a potential of success for the non-uniform atmosphere case must pass the uniform atmosphere test first. If one takes the position that the above (classical) sonic boom theory is essentially correct, then it appears that the route to achieve sonic boom reduction is through modification of f (X). See http://www.princeton.edu/∼mae/SHL/Heat.pdf to see a discussion on the wave drage generated by volume heat addition in a supersonic flow field and its cancellation by a “wake thrust” in the wake.

11

6

Epiloque

Everything I know on sonic boom theory was taught to me by Professor Wallace D. Hayes in his lengendary course 523, 524 (Fundamentals of Gas Dynamics and Advanced Gas Dynamics) in 1954-1955. I recall the elegance and pride of authorship of Wally’s lectures on the sonic boom theory, including his famed “equivalent bodies of revolution,” the concept that for a whole aircraft, the radiation (and wave drag) at each azimuthal angle θ can be represented by an equivalent body of revolution, its radius of revolution is computed by integrating the sources along the intersection of the forward Mach cone from the point of observation with the actual aircraft itself. This concept was later experimentally verified by Whitcomb, and was the theoretical foundation for the “coke bottle” shape of some supersonic fighters and bombers and the lumpy first class lounge of the Boeing 747. J. M. Burgers came to Princeton in either 1955 or 1956 to give a Baetjer Seminar on his recent work on what is now called the Burger’s equation (the application in his mind was turbulence). I vividly recall the vivid childish glee on his face when he presented the “equal area” algorithm for locating emerging discontinuities. Wally passed away peacefully in March, 2001. I dedicate these notes to him.

12