Untitled - Aerobib - Universidad Politécnica de Madrid

242 CHAPTER 9. FLOWS WITH COMBUSTION WAVES
9.6 Vorticity across the flame
In the flow of a perfect gas, where viscosity and mass forces are neglected, the variation
of the rotational ¯ω = ∇ × ¯v as a function of the stagnation temperature T s and
specific entropy S can be expressed as follows
∂ ¯ω
∂t + ¯v × ¯ω = c p∇T s − T ∇S. (9.54)
For isoenergetic (T s = const.) and stationary (∂/∂t = 0) flows, only this
case will be considered in the present study, 5 equation (9.54) reduces to the following
(Crocco’s Theorem)
¯v × ¯ω = −T ∇S. (9.55)
From this equation, the jump of the rotational, across the flame, can be computed
when the values of ¯v, T and S are known. The jump of the rotational is obtained by
(9.55) to both sides of the flame. For the calculation, we shall adopt on each point
of the flame front a cartesian rectangular coordinate system, defined in the following
way:
Axis n: Normal to the flame front.
Axis t: Intersection of the plane tangent to the flame with the plane of the incident
and emergent velocities ¯v 1 and ¯v 2 .
Axis τ: Normal to the (n, t) plane forming a positive trihedron.
The velocity components relative to this system, will be v n , v t and 0. The
vorticity components ω n , ω t and ω τ . Therefore, those of the vector product ¯v × ¯ω will
be v t ω τ , −v n ω τ and (v n ω t − v t ω n ). Consequently, equation (9.55) breaks down into
the following three equations 6
v t ω τ = −T ∂S
∂n , (9.56)
v n ω τ =
T ∂S
∂t , (9.57)
v n ω t − v t ω n = −T ∂S
∂τ . (9.58)
The jump of the normal component of the vorticity ω n can be computed directly.
The above equations are not necessary for the computation. In fact, since v t
5 If the flow before the flame is isoenergetic and if the heat q released in the combustion is constant, as
will be assumed hereinafter, the flow after the flame is also isoenergetic, as results from (9.14).
6 The derivative ∂S/∂t in the tangential direction must not be confused with a time derivatives.