subsonic and sonic jet flows · 2019-12-23 · there is another formation of the problem by...

Subsonic and Sonic Jet Flows

Zhouping Xin

The Institute of Mathematical Sciences

The Chinese University of Hong Kong

Joint works with Chunpeng Wang

Workshop on Nonlinear PDEs and Related Topics

Institute of Mathematical SciencesNational University of Singapore

December 30, 2019

Zhouping Xin Subsonic and Sonic Jet Flows

I. Backgrounds and Known Resultsfor Subsonic and Sonic Jet Flows

I. Backgrounds and Known Results for Subsonic and Sonic Jet Flows

Aim

Aim: For a given finitely long nozzle and a given surroundingpressure, get a subsonic or sonic jet flow.

Subsonic jet flow: a free boundary problem of a uniformlyelliptic equationSonic jet flow: a free boundary problem of a degenerateelliptic equation


Compressible Euler system

Consider the two dimensional compressible Euler system of steadyisentropic and irrotational flows

∂x(ρu) + ∂y (ρv) = 0,

∂x(P + ρu2) + ∂y (ρuv) = 0,

∂x(ρuv) + ∂y (P + ρv 2) = 0,

∂yu = ∂xv ,

P(ρ) =1

γργ ,

where (u, v), P and ρ represent the velocity, pressure and densityof the flow, respectively, and γ > 1 is the adiabatic exponent for apolytropic gas.


Full potential equation

For irrotational flows, the Euler system is transformed into the fullpotential equation

div(ρ(|∇ϕ|2)∇ϕ) = 0,

where

ρ(q2) =(

1− γ − 1

2q2)1/(γ−1)

, 0 < q2 <2

γ − 1,

andϕx = u, ϕy = v .

The sound speed c is defined as

c2 = P ′(ρ) = ργ−1 = 1− γ − 1

2|∇ϕ|2.


Full potential equation

div(ρ(|∇ϕ|2)∇ϕ) = 0

The full potential equation is elliptic in subsonic region(|∇ϕ| < c), degenerate at sonic state (|∇ϕ| = c), while hyperbolicin supersonic region (|∇ϕ| > c).

At the sonic state, the sound speed is

c∗ =( 2

γ + 1

)1/2,

which is critical.

The flow is subsonic when |∇ϕ| < c∗, sonic when |∇ϕ| = c∗ andsupersonic when |∇ϕ| > c∗.


Chaplygin equations

The full potential equation can be reduced to the the Chaplyginequations

∂θ

∂ψ+ρ(q2) + 2q2ρ′(q2)

qρ2(q2)

∂q

∂ϕ= 0,

1

q

∂q

∂ψ− 1

ρ(q2)

∂θ

∂ϕ= 0

in the potential-stream coordinates (ϕ,ψ), where q is the speed,while θ, called a flow angle, is the angle of the velocity inclinationto the x-axis.

Also the linear Chaplygin equation in the velocity coordinates

∂

∂q

( q

ρ(q2)

∂ψ

∂q

)+

1−M2

qρ(q2)

∂2ψ

∂θ2= 0,

where M =q

cis the Mach number.


Chaplygin equations in the potential-stream coordinates

Eliminating θ yields the following second-order quasilinear equation

∂2A(q)

∂ϕ2+∂2B(q)

∂ψ2= 0,

where

A(q) =

∫ q

c∗

ρ(s2) + 2s2ρ′(s2)

sρ2(s2)ds, B(q) =

∫ q

c∗

ρ(s2)

sds,

A′(q)

> 0, if 0 < q < c∗,

= 0, if q = c∗,

< 0, if c∗ < q <√

2/(γ − 1),

B ′(q) > 0, 0 < q <√

2/(γ − 1).


Known results: examples and numerical results

Problems of subsonic jet flows describe the elements ofhydrodynamics and have been extensively studied. Many examplesand numerical results can be found in the monographs.

S. Chaplygin, Gas jets, Tech. Memos. Nat. Adv. Comm.Aeronaut., 1944(1063)(1944), 112pp.

G. Birkhoff and E. H. Zarantonello, Jets, wakes, and cavities,Academic Press Inc., Publishers, New York, 1957.

L. Bers, Mathematical aspects of subsonic and transonic gasdynamics, John Wiley & Sons, Inc., New York; Chapman &Hall, Ltd., London, 1958.

D. Gilbarg, Jets and cavities, Handbuch der Physik, Vol. 9,Part 3, 311–445, Springer-Verlag, Berlin, 1960.


Known results: examples

S. Chaplygin, Gas jets, Tech. Memos. Nat. Adv. Comm.Aeronaut., 1944(1063)(1944), 112pp.

∂

∂q

( q

ρ(q2)

∂ψ

∂q

)+

1−M2

qρ(q2)

∂2ψ

∂θ2= 0.

There is a solution, as a power series in the velocity coordinates(q, θ). This solution solves the problem of the subsonic or sonic jetflow out of the opening vessel with plane walls.


Known results: examples of sonic jet

If the surrounding pressure corresponds the sonic state, this flowbetween the two sonic jets is sonic at a finite distance from thevessel, more precisely along a straight line. And beyond thisstraight line the flow is a uniform sonic flow.

L. V. Ovsiannikov, Gas flow with straight transition line(Russian), Akad. Nauk SSSR. Prikl. Mat. Meh., 13(1949).537–542, also Tech. Memos. Nat. Adv. Comm. Aeronaut.,1295(1951), 13 pp.


Known results: rigorous mathematical theory

In 1981–1985, H. W. Alt, L. A. Caffarelli and A. Friedmandeveloped a variational approach to solve subsonic jet flows, suchas jets from general nozzles, jets with gravity and jets with twofluids. There are also a series of recent works by L. L. Du and hiscooperators.Variational approach by Alt, Caffarelli and Friedman: which arebased stream function formulations


Main result by Alt, Caffarelli and Friedman

Theorem

For an infinitely long nozzle, if the mass flux of the flow isprescribed, then there is uniquely a subsonic jet flow which isinfinitely long after the nozzle and whose pressure on the freeboundary is a constant.


Main result by Alt, Caffarelli and Friedman

For given mass flux, the pressure of the subsonic jet flow onthe free boundary is a constant which is solved in theproblem. That is to say, the surrounding pressure cannon begiven in advance.

If the nozzle is finitely long, the boundary condition at theinlet is not clear.

For sonic jet, the variational method fail.


Open question I

Question

How to formulate a subsonic jet flow whose pressure on the freeboundary coincides with the given surrounding pressure?

If the surrounding pressure is given, the free boundary of asubsonic jet flow may be located in a bounded domain. Howto prescribe the boundary condition on the free boundary ofthe subsonic jet flow except that the pressure of the flow onthe free boundary coincides with the given surroundingpressure?

What other physical boundary conditions should be prescribedat the inlet?


Open question II

Question

For a given surrounding pressure, how to get a subsonic jet flowwhich is finitely or infinitely long? Is there a sonic jet flow?

♦ The sonic jet problem is a free boundary problem of adegenerate elliptic equation.

♦ Where are the sonic points and what is the regularity of theflow?


II. Subsonic and Sonic Jet Flowsfrom Convergent Nozzles

II. Subsonic and Sonic Jet Flows from Convergent Nozzles

Aim

For a given surrounding pressure, get a subsonic or sonic jetflow which is finite or infinitely long.


Aim

Jet Problem: For given positive constants R0 > 0, R ∈ (0,R0),

ϑ ∈ (0,π

2), and Pe ∈ [P(ρ(e2

∗)),1

γ]. We seek a subsonic (or sonic)

jet flow in Ω such that

(i) the incoming mass flux at Γin is a constant m;

(ii) the flow velocity at Γin and Γout is alone the normal direction;

(iii) the flow satisfies the slip condition on the wall Γw andstreamline free boundary Γws ;

(iv) the pressure of the flow at the free boundary Γws ∪ Γout is thegiven Pe ;

(v) in the case that the jet is infinitely long, there is no Γout .


Formulation in the physical plane

Fix R0 > 0, R ∈ (0,R0), ϑ ∈ (0, π/2),Pe = P(ρ(c2

e )) ∈ [P(ρ(c2∗ )), 1/γ) (or ce ∈ (0, c∗]).

♦ sonic jet if ce = c∗. Degeneracy causes an essential difficulty.


Formulation in the physical plane

Let c` ∈ (0, ce) be the unique solution to

ρ(c2` )(A(ce)− A(c`)) = 1

SetR∗ = R0 c`ρ(c2

` )/(ceρ(c2e ))

The solutions spaces will be

Cbd = (ϕ,Ωbd) : ϕ ∈ c1(Ωbd), c` ≤ |∇ϕ| < c∗ in Ωbd

C = (ϕ,Ω) : ϕ ∈ c1(Ω), c` ≤ |∇ϕ| ≤ c∗ in Ω


Formulation in the physical plane: finitely long jet

div(ρ(|∇ϕ|2)∇ϕ) = 0, (x , y) ∈ Ω, (1)

−∫

Γin

ρ(|∇ϕ(x , y)|2)∇ϕ(x , y) · ν(x , y)dl = m (2)

ϕ(x , y) = 0, (x , y) ∈ Γin, (3)

∇ϕ(x , y) · ν(x , y) = 0, (x , y) ∈ Γw ∪ Γws ∪ (x-axis ∩ ∂Ω), (4)

P(ρ(|∇ϕ(x , y)|2)) = Pe , (x , y) ∈ Γws ∪ Γout, (5)

ϕ(x , y) = ξ, (x , y) ∈ Γout, (6)

where m is a given constant, (ϕ,Ω) is an unknown solution,Γws ∪ Γout is a free boundary, ξ is a free constant.


Formulation in the physical plane: infinitely long jet

div(ρ(|∇ϕ|2)∇ϕ) = 0, (x , y) ∈ Ω, (7)

−∫

Γin

ρ(|∇ϕ(x , y)|2)∇ϕ(x , y) · ν(x , y)dl = m, (8)

ϕ(x , y) = 0, (x , y) ∈ Γin, (9)

∇ϕ(x , y) · ν(x , y) = 0, (x , y) ∈ Γw ∪ Γws ∪ (x-axis ∩ ∂Ω), (10)

P(ρ(|∇ϕ(x , y)|2)) = Pe , (x , y) ∈ Γws, (11)

where (ϕ,Ω,m) is an unknown solution, Γws is a free boundary, mis a free constant.

Remark

There is another formation of the problem by Wang-Xin, where Pe

and m are given, but R is regarded as a variable. However, thisformulation cannot deal with the infinitely long jet problem.


Well-posedness

Main ResultsFirst, we show that R ∈ (R∗,R0) is necessary in general forwell-posedness for both the finite and infinite jet problems.

Theorem (1)

Assume that R0 > 0, Pe ∈ [P(ρ(c2∗ )),

1

γ), θ ∈ (0,

π

2].

For R = R∗, the problem (1)-(6) has a unique solution in Cbd

if m = R∗θceρ(c2e ), (in this case, Γws = φ and the solution is

radially symmetric), while there exists no solution in Cbd ifm ∈ (0,R∗θceρ(c2

e )) ∪ (R∗θceρ(c2e ),+∞).

For R < R∗ and m > 0, @ solution to (1)-(6) in Cbd ;

For R < R∗, @ solution to (7)-(11) in C .


Well-posedness

Next result concerns the well-posedness of finitely long jet:

Theorem (2) (finitely long jet)

Assume that R0 > 0, Pe ∈ [P(ρ(c2∗ )), 1/γ), R ∈ (R∗,R0), and

ϑ ∈ (0, ϑ∗] ∩ (0, π/2) with θ∗ depending only on γ, R0, Pe and R.Then there exist two positive constants m∗ < m∗ depending onlyon γ, R0, Pe , R, and θ such that the problem (1)-(6) has auniquely solution in Cbd if and only if m ∈ (m∗,m

∗] for subsonicjet and m ∈ [m∗,m

∗] for sonic jet. Furthermore, m∗ = Rθceρ(c2e ),

and the flow is radially symmetric with Γws = φ when m = m∗.


Well-posedness

For the finitely long sonic jet flow problem, there is no sonicpoints in the domain.

The case that m = m∗ is not considered yet. It corresponds tothe infinitely long jet flow.

Theorem (3) (infinitely long jet)

The problem (7)–(11) admits uniquely a solution (ϕ∗,Ω,m) with(ϕ,Ω) ∈ C and m = m∗.


Relation between the finitely and the infinitely long jets

♦ Finitely long jet: existence and uniqueness if m ∈ (m∗,m∗],

nonexistence if m ∈ (0,m∗) ∪ (m∗,+∞).♦ Infinitely long jet: there is a unique solution (ϕ∗,Ω∗,m∗).

Theorem (4)

X m = m∗ (the largest mass), the flow is symmetric (Γws = ∅).X m = m∗ (the smallest mass), the jet is infinitely long (Γout = ∅).


Regularity of the finitely long subsonic jet

Let ϕ be the subsonic jet flow in Theorem (1), m ∈ (m∗,m∗].

Theorem (5) (finitely long subsonic jet)

♦ Regularity: ϕ ∈ C 1,α(Ωm) and Γw ∪ Γws ∈ C 1+α for eachexponent α ∈ (0, 1/2) (Optimal in the sense that α = 1/2 for theLaplacian equation). Γws \ (Γw ∩ Γws) ∈ C 1,1, Γout ∈ C 1,1.♦ Geometry of the free boundary: both Γws and Γout are strictlyconvex, whose tangent lines are located on the same side as theflow.


Regularity of the finitely long sonic jet

For the finitely long sonic jet in Theorem 1,

Theorem (6) (finitely long sonic jet)

♦ Regularity: ϕ ∈ C 1(Ωm), Γw ∪ Γws ∈ C 1,Γws \ (Γw ∩ Γws) ∈ C 1,1, Γout ∈ C 1,1.♦ Geometry of the free boundary: Γws is strictly convex, whosetangent lines are located on the same side as the flow. Form ∈ (m∗,m

∗], Γout is convex, and the flow angle is non-zero onΓout except for the point on the x-axis. For m = m∗, Γout is a linesegment parallel to the y-axis.♦ Singularity: for m ∈ (m∗,m

∗], the acceleration blows up on Γout,and the flow cannot be extended to be a global one. For m = m∗,no singularity, the flow can be extended globally by a sonicextension.


Regularity of the infinitely long subsonic jet

Let (ϕ∗,Ω∗,m∗) be the infinitely long subsonic jet in Theorem (1).

Theorem (7) (infinitely long subsonic jet)

♦ Regularity: ϕ∗ ∈ C 1,α(Ω∗) and Γw ∪ Γws ∈ C 1+α for eachexponent α ∈ (0, 1/2) (Optimal in the sense that α = 1/2 for theLaplacian equation). Γws \ (Γw ∩ Γws) ∈ C∞.♦ Geometry of the free boundary: Γws is strictly convex, whosetangent lines are located on the same side as the flow, |∇ϕ∗| < cein Ω∗.


Regularity of the infinitely long sonic jet

Let (ϕ∗,Ω∗,m∗) be the infinitely long sonic jet in Theorem (1).

Theorem (8) (infinitely long sonic jet)

♦ Regularity: ϕ∗ ∈ C 1(Ω∗), and Γw ∪ Γws ∈ C 1,Γws \ (Γw ∩ Γws) ∈ C 1,1.♦ Geometry of the free boundary: there exists x < −R such that

|∇ϕ∗(x , y)|

< c∗, if (x , y) ∈ Ω∗, x < x

= c∗, if (x , y) ∈ Ω∗, x ≥ x ,

∇ϕ∗(x , y) = (c∗, 0), (x , y) ∈ Ω∗, x ≥ x .

Γws is strictly convex on the left-hand side of x = x , whose tangentlines are located on the same side as the flow, while Γws is astraight line parallel to the x-axis on the right-hand side of x = x .


Infinitely long sonic jet flows

-

6

x

y

Γw

Γin

Γw

Pe = P(c2∗ )

Pe = P(c2∗ )


III. Some Key Steps in the Analysis


Formulation in the potential plane: finitely long jet

We start with finitely long jet problem.

∂2A(q)

∂ϕ2+∂2B(q)

∂ψ2= 0, (ϕ,ψ) ∈ (0, ξ)× (0,m), (12)

∂A(q)

∂ϕ(0, ψ) =

1

R0q(0, ψ)ρ(q2(0, ψ)), ψ ∈ (0,m), (13)

∂B(q)

∂ψ(ϕ, 0) = 0, ϕ ∈ (0, ξ), (14)

∂B(q)

∂ψ(ϕ,m) = 0, ϕ ∈ (0, ζ), (15)


Formulation in the potential plane: finitely long jet

q(ϕ,m) = ce , ϕ ∈ (ζ, ξ), (16)

q(ξ, ψ) = ce , ψ ∈ (0,m), (17)

m = R0ϑMρ(M2), (18)∫ m

0

1

q(0, ψ)ρ(q2(0, ψ))dψ = R0ϑ, (19)

where ζ > 0 and M ∈ (0, c∗) are given constants, and (q, ξ) is anunknown solution with ξ ≥ ζ being a free constant.


Formulation in the potential plane: infinitely long jet

∂2A(q)

∂ϕ2+∂2B(q)

∂ψ2= 0, (ϕ,ψ) ∈ (0,+∞)× (0,m), (20)

∂A(q)

∂ϕ(0, ψ) =

1

R0q(0, ψ)ρ(q2(0, ψ)), ψ ∈ (0,m), (21)

∂B(q)

∂ψ(ϕ, 0) = 0, ϕ ∈ (0,+∞), (22)

∂B(q)

∂ψ(ϕ,m) = 0, ϕ ∈ (0, ζ), (23)

q(ϕ,m) = ce , ϕ ∈ (ζ,+∞), (24)

m = R0ϑMρ(M2), (25)∫ m

0

1

q(0, ψ)ρ(q2(0, ψ))dψ = R0ϑ, (26)

where ζ > 0 is a given constant, (q,M) is an unknown solutionwith M ∈ (0, c∗) being a free constant.


Difficulties in the potential plane

Solve the problem in the potential plane.

Difficulties for the finitely long subsonic jet flow problem:

At the inlet the flow satisfies a Robin boundary condition withwhich even the fixed boundary problem may be ill-posed.

The problem is not a perturbed problem and there is nobackground solution.

It is mixed Dirichlet-Neumann boundary conditions on astreamline.

New Difficulties:

Unbounded domain, new global estimates.

The equation is degenerate at sonic points and the location ofthe sonic points is unknown.

Where are the sonic points and what is the regularity of theflow?


Location of sonic points for subsonic-sonic flows

Theorem (D. Gilbarg and M. Shiffman, 1954, smooth flows)

Let G be a domain in the physical plane. Consider a C 2(G )subsonic-sonic flow. If the flow is sonic at a point P ∈ G . Then,the flow is sonic on the longest open segment lying wholly in Gwhich contains P and is vertical to the velocity of the flow at P.

Theorem (Wang & Xin, 2017, continuous flows)

The above theorem still holds for a C (G ) subsonic-sonic flow.

qPHHY(u, v)

subsonic

subsonic


Location of sonic points for subsonic-sonic flows

Theorem (Wang & Xin, 2019)

Assume that 0 < ζ < ξ, q ∈ C ([0, ξ]× [0,m]) is a solution to theproblem (12)–(17) with ce = c∗. Then there exists a numberξ ∈ (ζ, ξ] such that

q(ϕ,ψ)

< c∗, if (ϕ,ψ) ∈ [ζ, ξ)× [0,m),

= c∗, if (ϕ,ψ) ∈ [ξ, ξ]× [0,m].

Theorem (Wang & Xin, 2019)

Assume that q ∈ C ([0,+∞)× [0,m]) is a solution to the problem(20)–(24) with ce = c∗. Then there exists a number ξ ∈ (ζ,+∞)such that

q(ϕ,ψ)

< c∗, if (ϕ,ψ) ∈ [ζ, ξ)× [0,m),

= c∗, if (ϕ,ψ) ∈ [ξ,+∞)× [0,m].


Length of the nozzle

solutions : inf q ≥ cl

The length of the nozzle satisfies

R0clρ(c2l )

(ceρ(c2e ))≤ R < R0.

(For the longest nozzle, there is only symmetric flow.)

In potential plane, it is 0 < ζ ≤ R0cl .


Condition for uniqueness of the fixed boundary problem

The uniqueness of the fixed boundary problems (12)–(17) and(20)–(24) is proved by a duality argument.

a(ϕ,ψ)∂2V

∂ϕ2+∂2V

∂ψ2= 0, (ϕ,ψ) ∈ (0, ξ)× (0,m),

∂V

∂ϕ(0, ψ) = − 1

R0cl, ψ ∈ (0,m),

∂V

∂ψ(ϕ, 0) = 0, ϕ ∈ (0, ξ),

∂V

∂ψ(ϕ,m) = 0, ϕ ∈ (0, ζ),

V (ϕ,m) = 0, ϕ ∈ (ζ, ξ),

V (ξ, ψ) = 0, ψ ∈ (0,m)

admits a solution satisfying sup(0,ξ)×(0,m)

V < 1, where

a ∈ C ([0, ξ]× [0,m]) and 0 ≤ a ≤ a0.X Degenerate for sonic jet flows. Need modification.


Condition for uniqueness of the fixed boundary problem

To get sup(0,ξ)×(0,m)

V < 1, one should restrict an upper bound of m.

Proposition

There exists a positive constant κ1 > 0 depending only on γ, ceand ζ such that(i) If 0 < m ≤ κ1R0, then for each ξ ≥ ζ and eacha ∈ C ([0, ξ]× [0,m]) (0 < a ≤ a0 in (0, ξ)× (0,m)),

sup(0,ξ)×(0,m)

V < 1.

(ii) If m > κ1R0, then there exist ξ ≥ ζ anda ∈ C ([0, ξ]× [0,m]) (0 < a ≤ a0 in (0, ξ)× (0,m)) such that

sup(0,ξ)×(0,m)

V > 1.


Condition for existence of the fixed boundary problem

For the existence, one should restrict an upper bound of m.

Proposition

There exists a positive constant κ2 > 0 depending only on γ, ceand ζ such that(i) If 0 < m ≤ κ2R0, then for each ξ ≥ ζ, the problem (12)–(17)admits a solution satisfying inf

(0,ξ)×(0,m)q ≥ cl .

(ii) If m > κ2R0, then there exists ξ ≥ ζ such that there is not asolution to the problem (12)–(17) satisfying inf

(0,ξ)×(0,m)q ≥ cl .

Proposition

(i) If 0 < m ≤ κ2R0, then the problem (20)–(24) admits a solutionsatisfying inf

(0,+∞)×(0,m)q ≥ cl .

(ii) If m > κ2R0, then there is not a solution to the problem(20)–(24) satisfying inf

(0,+∞)×(0,m)q ≥ cl .


Well-posedness of the fixed boundary problems

Theorem

For R0 > 0, ce ∈ (0, c∗], ζ ∈ (0,R0cl), 0 < m ≤ minκ1, κ2R0

and ξ ≥ ζ, the problem (12)–(17) admits a unique solutionq ∈ C ([0, ξ]× [0,m]) with inf

(0,ξ)×(0,m)q ≥ cl .

Theorem

For R0 > 0, ce ∈ (0, c∗], ζ ∈ (0,R0cl) and0 < m ≤ minκ1, κ2R0, the problem (20)–(24) admits a uniquesolution q ∈ C ([0,+∞)× [0,m]) with inf

(0,+∞)×(0,m)q ≥ cl .

Remark

For the sonic jet flow problem, the equation is degenerate. Thewell-posedness of the continuous solution is proved. The proof isdifficult.


Well-posedness of the fixed boundary problems

We need some global estimates for the problems inunbounded domains (V < 1, q ≥ cl).

We establish the well-posedness for the continuous solution.The properties of the continuous solution are also shown, suchas regularity, monotonicity, asymptotic behavior, continuousdependence with respect to ζ, ξ and m.

The equation is degenerate at sonic points and the location ofsonic points is unknown. Some new techniques to study thelocation of sonic points and the geometry of the sonic sethave been used. The asymptotic behavior at the boundary ofthe sonic set is shown.


Upper bound of the nozzle angle

For R0 > 0, Pe ∈ [P(ρ(c2∗ )), 1/γ), R ∈ (R0clρ(c2

l )/(ceρ(c2e )),R0),

the mass flux satisfies

0 < m = R0ϑMρ(M2) ≤ minκ1, κ2R0,

⇑

ϑ ≤ ϑ∗ =minκ1, κ2

ceρ(c2e )

, (ϑ∗ depending only on γ and ce).


Many Thanks!


subsonic and sonic jet flows · 2019-12-23 · there is another formation of the problem by...

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