Chapter 1 Introduction
2.2 Euler equations for Compressible Flow
The 3D Euler equations are a nonlinear hyperbolic system and consist of the conti- nuity, momentum, and energy equations. The continuity equation is in differential form
ρ,t+ (ρui),i= 0, (2.2)
where ρ is the density and ui represents the velocity in the x, y, z directions. The momentum equation is
(ρuj),t+ (ρuiuj),i =−p,j, (2.3) where p is the pressure with P,j = P,iδij. Unlike the scalar continuity and energy equations, the momentum equation is a vector equation with one equation for each of the three spatial dimensions. The last equation, the energy equation, is expressed as
(ρet),t+ (ρuiet),i =−(uip),i, (2.4) where et is the total internal energy. Now, for a compact representation, one uses a vector equation, that represents a system of 5 scalar equations. The final result is a nonlinear, time dependent, hyperbolic, vector equation. This 5 degree of freedom equation is written as
U,t+Fadvi,i = 0. (2.5)
With
U=
ρ ρu1 ρu2
ρu3 ρet
, Fadvi =uiU+p
0 δ11 δ22
δ33 ui
. (2.6)
2.2.1 Inviscid, Adiabatic, One-Dimensional Similarity Flow
Inviscid, adiabatic, one dimensional flow as described by the Euler equations has no characteristic length or time scales present. Only a velocity scale dictates the form of the solution. In this case, all derivatives with respect to x and t can be expressed
in terms of the similarity parameter ξ, see (94)§99,
ξ=x/t, (2.7)
∂
∂x = 1 t
d
dξ, (2.8)
∂
∂t =−ξ t
d
dξ. (2.9)
These expressions can be applied to the continuity and Euler equations
∂ρ
∂t + ∂(ρv)
∂x = 0, (2.10)
∂v
∂t +v∂(v)
∂x =−1 ρ
∂p
∂x, (2.11)
with v as the x component of the velocity. Now writing all derivatives in terms of ξ, the t and xvariables disappear.
(v−ξ)dρ
dξ +ρd(v)
dξ = 0, (2.12)
(v−ξ)dv
dξ =−1 ρ
dp
dξ. (2.13)
By expressing the equations in terms of the similarity variable, exact solutions can be found, for example for the breaking wave problem, see appendix A. These exact solutions have previously been used for verification of the numerical methods for the inviscid part of the solutions. Understanding similarity flow is also crucial for understanding the physics of shock reflections.
2.2.2 Shock Reflections
Shock waves are a fundamental property of the Euler equations and hyperbolic sys- tems in general. To understand supersonic compressible flows, one must study shock wave interactions with solid boundaries and other shocks.
The following section summarizes the basic theory for modeling the interactions of planar shock waves with angled wedges, particularly the properties of the triple point structure. This problem is important for understanding how shock waves interact with boundaries and with other shock waves. When a moving planar shock wave hits a wedge, the shock is reflected and the flow is deflected. This is referred to as the
“unsteady problem”. The incident shock wave is moving unchanged, encounters the
wedge, and then moves up the wedge. Unsteady examples are shown in figures2.1(a), and 2.1(b). For the “steady problem” a stationary shock wave at an oblique angle encounters a boundary and reflects off. An example of where this may occur is in a jet engine inlet or outlet, or in shock tubes with traverse waves. Steady examples are shown in Fig. 2.1(c) and 2.1(d). The unsteady and steady cases are different, however, they have inherently similar fundamental processes involved.
i
r
θ w
(a) Pseudo-steady regular re- flection
i
r m T s
θ w χ
(b) Pseudo-steady Mach re- flection
i r
θ w
M0
1 2
(c) Steady regular reflection
i r
m T s
θ w
M0
0 1
2 3
(d) Steady Mach reflec- tion
Figure 2.1: Example shock reflections
For both the steady and unsteady cases, the Mach number and the wedge an- gle are the input parameters for a perfect gas model. For the real gas model, the pressure, temperature, and species data are also required. There also exist different levels of approximation for modeling chemical reactions. In all cases of inviscid the- ory,thermodynamic equilibriumis enforced across all shock waves. The chemicalpost shock state, for example, from dissociation, combustion, or other processes, can be either ignored; (frozen case) as in the case of relatively low temperatures of nonreact- ing species; assumed to be in local equilibrium (all reaction occur across shock); or assumed to be partially reacted, by simulating the unsteady chemical kinetics.
Approximate solutions known as “two shock theory” and “three shock theory” for the shock reflection phenomenon were developed by (156). The three shock theory models the primary triple point shock structure of the more complex reflection phe- nomenon known as Mach reflection. For the unsteady problem, the triple point path
angle is found as a function of the incident shock Mach number and the wedge angle.
These results agree roughly with experimental results, except at small wedge angles and for strong Mach numbers where the assumptions break down the most (16).
For further developments and examples from these shock theories and a discussion of the various types of shock reflections see Appendices B and C.
2.2.2.1 Self Similar Approximation
The unsteady reflection process can be approximated as a self similar, pseudo-steady problem (103), (15), (18). For this simplification, the Mach stem is assumed to be straight and perpendicular to the wedge, and the fluid is modeled as inviscid.
By attaching a reference frame to the triple point, the three shock pseudo-steady solution is found. In this frame, the reflected wave of a single Mach reflection (SMR) relative to the triple point path, is straight in the pseudo-steady reference frame. This is demonstrated in figure 2.2, where the velocity vectors are shown in the reference frame of the triple point.
Figure 2.2: Velocity vectors in the triple point reference frame, DMR case