The Reference Temperature/Enthalpy Method for Compressible Boundary Layers

In [14] it is shown that for laminar high-speed flows boundary-layer skin fric­tion and wall heat transfer can be obtained with good accuracy by employing the relations for incompressible flow, if the fluid property density p and the transport property viscosity p are determined at a suitable reference temper­ature T *. The approach is based on the observation that the results of the investigated boundary-layer methods depend strongly on the exponent ш of the employed viscosity relation, see Section 4.2, on the wall temperature Tw, and on the boundary-layer edge Mach number Me, which is representative for the ratio ‘boundary-layer edge temperature’ to ‘total temperature of the flow’, Te/Tt. The dependence on the Prandtl number Pr is weak.

The reference temperature approach is interesting, although the complex interactions of convective and molecular heat transfer, compression and dis­sipation work, see eq. (4.58) or (7.40), with the boundary conditions at the body surface and at the boundary-layer’s outer edge do not suggest it at a first glance. We will see later, that theory based on the Lees-Dorodnitsyn transformation to a certain degree supports this approach.

The reference temperature concept was extended to the reference enthalpy concept [15] in order to take high-temperature real-gas effects into account. In [15] it is shown that it can be applied with good results to turbulent boundary layers, too.

Reference temperature (T*) for perfect gas and reference enthalpy (h*) for high-temperature real gas, respectively, combine the actual values of the temperature T or enthalpy h of the gas at the boundary layer’s outer edge (e), at the wall (w), and the recovery value (r) in the following way:

T * =0.28Te + 0.5Tw +0.22Tr, (7.62)

h* =0.28he + 0.5hw + 0.22hr. (7.63)

Eq. (7.63) contains eq. (7.62) for perfect gas.

Because T*, respectively h*, depend on boundary-layer edge data and in particular on wall data, we note

on general vehicle surfaces : T* = T*(x, z), resp. h* = h*(x, z),

i. e., reference temperature or enthalpy are not constant on a vehicle surface, especially if this surface is radiation cooled.

The recovery values are found in terms of the boundary-layer edge data:

V2 Tr = Te+r*-Ц 2cp	(7.64)
V2 hT = he+rl-f.	(7.65)
Eq. (7.64) can also be written as
Tr = Te(l + r*^^-M;).	(7.66)
The recovery factor in these relations is
r* = rf = / Pr*,	(7.67)

for laminar flow, and with acceptable accuracy

r* = r*h = VPr*, (7.68)

for turbulent flow, with Pr* being the Prandtl number at reference-tempe­rature conditions:

P c*

Pr* = (7.69)

With these definitions the reference temperature, eq. (7.62), becomes in terms of the boundary-layer edge Mach number Me

T* = 0.5Te + 0.5TW + 0.22r*le~l M;Te. (7.70)

Eqs. (7.62) and (7.63) were found with the help of comparisons of results from solutions of the boundary-layer equations with data from experiments [14, 15]. They are valid for air, and for both laminar and turbulent boundary layers. Other combinations have been proposed, for instance, for boundary layers at swept leading edges, Sub-Section 7.2.4.

We employ now and in the following sub-sections only the reference tem­perature approach to demonstrate thermal surface effects in connection with attached viscous flow. This is permitted, because today materials of thermal protection systems or hot primary structures allow temperatures at most up to 1,800-2,000 K, see the examples given in Section 5.6 (RV’s) and in Sec­tion 7.3 (CAV). The power-law relations given in Section 4.2 for the viscosity and the thermal conductivity can be strained up to these temperatures. Vi­brational excitation, Chapter 5, possibly can be neglected. This all holds for qualitative considerations. For quantitative considerations one anyway has to establish first whether an approximate relation can be used for a given problem and how large the immanent errors are.

We use now ‘ *’ to mark reference-temperature data and relate them to overall reference flow parameters, which we mark with ‘ref ’. For the Reynolds number Re*x we thus find

4= P VrefX ___ PrefVrefX p Pref ______ p P Pref

^ex ~ I ~ 7~ ~ xi-eref, x —

P Pref Pref P Pref P

In attached viscous flow the pressure to first order is constant through the boundary layer in direction normal to the surface, and hence we have locally in the boundary layer:

P*T* = Pref Tref.

We introduce this into eq. (7.71) together with the approximate relation p = cTu for the viscosity, Sub-Section 4.2, and finally obtain[92]

T 1+ш

Ret = Reref}X ( /./ j. (7.73)

For flat plates at zero angle of attack, and approximately for slender bodies at small angle of attack, except for the blunt nose region, we can choose ‘ref’ = ‘TO’, whereas in general the conditions at the outer edge of the boundary layer are the reference conditions: ‘ref’ = ‘e’.

The reference-temperature/enthalpy extension of incompressible boun­dary-layer relations is not only a simple and effective method to demonstrate thermal-surface effects on attached viscous flow. In its generalized formula­tion given by G. Simeonides, [16], it is also an effective tool for the actual determination—with sufficient accuracy—of properties of attached compress­ible laminar or turbulent viscous flows, even for flows with appreciable high – temperature thermo-chemical effects, see, e. g., [17]. In Sub-Section 7.2.1 we give the generalized relations for the determination of boundary-layer thick­nesses and integral parameters and in Sub-Sections 7.2.3 to 7.2.6 for the determination of skin friction and thermal state of the surface.

The generalized relations can be applied on generic surfaces with either inviscid flow data found from impact methods or in combination with invis­cid flow field data found by means of Euler solutions. Of course only weak three-dimensionality of the flow is permitted. The stream-wise pressure gra­dient in principle also must be weak, but examples in [17] show that flows with considerable pressure gradients can be treated with good results. Flow separation and re-attachment, see Section 9.1, strong interaction phenomena and hypersonic viscous interaction, see Sections 9.2 and 9.3, must be absent, also slip-flow, Section 9.4. To describe such phenomena Navier-Stokes/RANS methods must be employed. Slip flow, however, can also be treated in the frame of boundary-layer theory.