Sensitivity of Stationary Shocks Claude Bardos and Olivier Pironneau Dedicated to Pierre Perrier
Abstract Sensitivity of shocks to data is a key point for fluid-structure and flutter control. We show here on a simple example that the derivative of aerodynamic variables with shocks with respect to data are solutions of non-standard partial differential equations. We present also some constructive proofs of existence of solutions and a few simple numerical example. Keywords: Partial differential equations, nozzle flow, sensitivity, transonic equation.
Sensitivity of the position of the shocks with respect to the parameters of the flow is the problem we would like to investigate here. There are many important applications such as the fluttering of wings. Godlewski et al [4] have studied a similar situation for the shock tube flow problem and solved it completely for Burger’s equation when the sensitivity is with respect to initial data. Giles [7] showed that the adjoint equation of the time dependent Euler equation is well posed and continuous across the shock, but to our knowledge no control of the shock position has been tried with Giles’ conditions.
Universit´e Paris VII and Universit´e of Paris VI+IUF (
[email protected])
1
Here, following [9] and [10] we analyze the stationary problem for the transonic equation. We derive formally an equation for the derivative of the potential of the flow with respect to parameters in the data; we show that the problem is well posed and we present an alternative mixte formulation which could be extended to more complex situations. Finally we present some numerical test which confirm the theory but which are still too preliminary for real life applications.
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Consider the Euler equations for compressible perfect isentropic gas in a domain Ω : ∂- ρ + ∇ · (ρu) = 0,
∂- ρu + ∇ · (ρu ⊗ u) + ∇ρ. = 0.
For stationary flows, the following holds ∇ · (u ⊗ u) = u∇u = ∇(
u2 ) − ρu × ∇ × u 2
so that there are irrotational solutions to the stationary Euler equations which then reduce to ∇ · (ρu) = 0,
∇(
ρu2 γ + ρ. ) = 0, 2 γ−1
∇×u=0
The second equation gives an algebraic relation between ρ and u2 : γ ρ.0/ γ −1
1
=K−
u2 , 2
where the constant K is fixed by the boundary conditions. The third equation tells us that u derives from a potential φ, i.e. u = ∇φ. The first equation, which determines φ, is known as the transonic equation. After renormalization, the transonic equation in a domain Ω of boundary Γ = Γ1 ∪ Γ2 , reads: ∇ · (1 − |∇φ|2 )1 ∇φ = 0 in Ω,
2
ρ
∂φ |Γ = g, ∂n 1
φ|Γ2 = (1) φΓ
with γ = 1.4, β = 1/(γ − 1) = 2.5 in air. Boundary conditions for nozzle flow on Γ = ∂Ω are of two kinds ∂φ |Γ 2 = u 3 ∂n
φ− < φ >= φ 4
on Γ 4 = Γ\Γ (2) 3
for some averaging operator (this is because the problem must give a solution up to an undefined constant). Γ 3 must contain the lateral walls of the nozzle Γ 5 where the flow is tangent to the walls and so u3 = 0. It can also contain either Γ 6 is the inflow boundary and/or Γ 7 the outflow one. If Γ 3 = Γ then it is necessary to add a compatibility relation on the data and a condition on φ Z (1 − |∇φ|2 )1 ∇φ · n = 0 (3) Γ
An entropy inequality must be added for well posedness (see Glowinski [11] and Neˇcas [5] ): ∆φ > −∞. It is automatically satisfied when u is continuous and also when u is discontinuous with a decreasing jump in the direction of the flow u.
8
;: being the characteristic function of a set D, u = u + (u+ − u )IΩ+
/
/
where u = are smooth functions. So δu = δu +(δu+ −δu )IΩ+ −δa(u+ −u )·nΣ αnΣ δΣ
/
/
/
3
where δΣ is the Dirac function on Σ. This is because the derivative of I > is a Dirac mass on ∂D. So if φ? , u? denotes the derivative of φ, u with respect to a, then we expect that
? − u? )IΩ+ − [u.nΣ ]αnΣ δΣ u ? = u ? + (u+ /
/
and @BADC+EGFBtherefore HJIKALHJMJFNMOC if u ? has a Dirac mass on Σ, φ ? must be across Σ, and u ? being the x-derivative of φ? :
HQAR@0SHTIUALIKV
