Test Methods and Conditions
Chapter 4 Analysis
4.1 Hypersonic Boundary Layer Mean Flow
4.1.1 Similarity Solution
The planar, two-dimensional, steady boundary layer equations for a compressible fluid are derived by White (1991). Continuity:
∂
∂x(ρu) + ∂
∂y(ρv) = 0 x-momentum:
ρ
u∂u
∂x +v∂u
∂y
=−∂pe
∂x + ∂
∂y
μ∂u
∂y
1See Section 4.2 for a description of the eN method.
y-momentum:
∂p
∂y = 0 Energy:
ρ
u∂h
∂x +v∂h
∂y
=u∂pe
∂x + ∂
∂y
k∂T
∂y
+μ ∂u
∂y 2
Where the fluid enthalpy ish=e+p/ρ, and both temperature and enthalpy depend upon pressure and density only: T = T(p, ρ) and h =h(p, ρ). This formulation can be used for fully mixed gas mixtures, such as atmospheric air.
White (1991) Chapter 7 provides an expression for a similarity solution in com- pressible flow which uses the transformation of the arbitrary two-dimensional equa- tions above. Illingworth (1950), quoted in White, separated the effects of viscosity and density into similarity variables ξ and η, respectively, for a flat plate:
ξ(x) = x
0
ρe(x)Ue(x)μe(x)dx (4.1)
η(x, y) = Ue(x)
√2ξ y
0
ρdy (4.2)
Lees (1956) performed a transformation of this flat-plate solution for axisymmetric laminar compressible boundary layers on a cone with less than 55◦ half-angle, for which the shock wave remains attached to the vertex of the cone, and flow at the cone surface may be assumed to have constant temperature, velocity, and pressure:
ξ(x) = x
0
ρe(x)Ue(x)μe(x)r20j(x)dx (4.3)
η(x, y) = ρeUe(x)r0j(x)
√2ξ
y
0
ρ
ρedy (4.4)
Here j = 0 for flow over a flat plate and j = 1 for axisymmetric flow, and the
body surface radius is r0 =bx with b= arctan 5◦ for the laminar boundary layer of a 5◦ half-angle cone. This modification is analogous to the Mangler transformation for incompressible flow, as described in White (1991) Chapter 4. Equations 7-80 from White (1991)2 reproduce the Lees solution:
ξ= ρeUeμeb2x3
3 (4.5)
dηcone =
3Ue 2xρeμe
1/2
ρdy (4.6)
For the axisymmetric conical geometry, the following equations for momentum and energy, respectively, are given by White (1991), where f(η) = u/ue is a shape function normalized by the velocity at the boundary layer edge, and g(η) =h/he is a shape function normalized by enthalpy at the boundary layer edge. The right hand side of the second equation constitutes the viscous dissipation term. Primes represent differentiation by η (the variable accounting for density effects):
(Cf) +f f = 0 (4.7)
(Cg)+ Prf g =−PrC(γ −1) M2ef2 (4.8)
Here C is the Chapman-Rubesin parameter, defined as the ratio of the products of density and viscosity at a location within the boundary layer to the products of density and viscosity at the edge. If the boundary layer profile is indeed self-similar, then the Chapman-Rubesin parameter will also be a function of η:
2Equations 7-80 in the Second Edition of White (1991) are incorrect for dηcone. The relationship is stated correctly here.
C = ρμ
ρeμe ≈C(η)
The gas flow in the boundary layers of the present experiments is hot but at low pressure. This means that the gas can be treated as ideal (P =ρRT), but the compo- sition may not be constant due to chemical reaction and transport processes within the boundary layer. In general, the enthalpy is a function of both species and tem- perature, and vibrational nonequilibrium and relaxation within the boundary layer may be significant in some cases. These important issues cannot be treated within the framework of self-similar solutions and are addressed in the present study by use of numerical solutions of the boundary layer equations as discussed in Section 4.1.2.
Nevertheless, in many cases it is possible to neglect the effect of composition changes and vibrational nonequilibrium and use the self-similar solutions as an ap- proximation to the actual boundary layer flow. As first suggested by Chapman and Rubesin, the dependence of C(η) can be determined through correlation with other thermodynamic properties. The first step is to relate the density within the boundary layer to temperature, making the standard assumption of constant pressure across the boundary layer and applying the ideal gas relationship for a fixed composition. In- stead of temperature, it is made convenient to use enthalpy as a variable, motivated by the case of constant heat capacity, and density is further approximated with a power-law dependence on enthalpy:
ρe
ρ(y) = T(y)
Te = h(y) he
Therefore, assuming a power-law relationship between viscosity ratio and temperature ratio, C can be expressed in terms of the enthalpy ratio only:
C = ρ ρe
μ μe ≈ he
h T
Te n
≈ h
he n−1
=g(η)n−1
For air, n ≈ 2/3 and thus C ≈ g−1/3 (White, 1991). For non-zero wall tempera- ture, assuming constant pressure across the boundary layer, an ordinary differential equation with the following five boundary conditions may be defined:
From the no-slip condition:
f(0) = f(0) = 0 From free stream velocity:
f(∞) = 1 From free stream enthalpy:
g(∞) = 1 From constant wall temperature:
g(0) = CpwTw CpeTe
= Tw Te
This ODE may be solved numerically using (for example) theMatlabbvp4crou- tine. The boundary conditions at infinity are incorporated numerically by specifying them at a relatively large number forη, then confirming that using successively larger values of η for “infinity” does not change the solution. Figure 4.1(a) demonstrates this asymptotic effect for five values of η from 15 to 20. In each case, df /dη → 1 within machine precision.
The expression for dηcone in Equation (4.6) is numerically integrated to find the relationship betweenηandy, and solutions foru/Ue =f(y),T /Te=g(y), andρ/ρe= 1/g(y) are computed. Examples of these solutions at conditions corresponding to one T5 experiment in air, shot 2742 (hres= 8.64 MJ/kg, Pres= 55.7 MPa) are presented in Figure 4.1. Similarity solutions of this form are computationally inexpensive and useful as a guide to the general properties of the compressible boundary layer on a 5◦ half-angle cone. However, a more sophisticated approach, detailed in Section 4.1.2, is
necessary to model the nonsimilar and nonequilibrium properties.
0 5 10 15 20
0 0.2 0.4 0.6 0.8 1 1.2
η
df/dη
(a) Asymptotic behavior asη→ ∞
0 1 2 3 4 5
0 0.2 0.4 0.6 0.8
ρ/ρe
y (m)
x = 0.1m x = 0.3m x = 0.5m
(b) Density
0 0.5 1 1.5
0 0.2 0.4 0.6 0.8
T/Te
y [mm]
x = 0.1m x = 0.3m x = 0.5m
(c) Temperature
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8
u/Ue
y [mm]
x = 0.1 m x = 0.3 m x = 0.5 m
(d) Velocity
Figure 4.1: Similarity solution convergence and profiles for density, temperature, and velocity, normalized by the boundary layer edge conditions, at three x-locations along the surface of the cone. For clarity, since the five df /dη curves in (a) are nearly identical, a circle indicates theη at which the boundary conditions at “infinity” were imposed for each curve. As the curves do not vary with increasing η from 15 to 20, the boundary conditions have been enforced at a sufficiently large value of η.