The Thermal State of Flat Surface Portions
The thermal state of the surface governs thermal-surface effects on wall and near-wall viscous and thermo-chemical phenomena, as well as the thermal loads on the structure, Section 1.4. Important is the fact that external surfaces of hypersonic flight vehicles basically are radiation cooled. Above we have seen, how the thermal state of the surface influences via Tw the boundary – layer thicknesses, and hence also the wall-shear stress of laminar or turbulent flow. In view of viscous thermal-surface effects, the thermal state of the surface is of large importance for CAV’s. Thermo-chemical thermal-surface effects in particular are important for RV’s. Thermal loads finally are of large importance for all vehicle classes.
We have defined the thermal state of a surface by two entities, the temperature of the gas at the wall which in the continuum-flow regime is the wall temperature Tgw = Tw, and the temperature gradient dT/dygw in the gas at the wall. For perfect gas or a mixture of thermally perfect gases in equilibrium the latter can be replaced by the heat flux in the gas at the wall
In Section 3.1 several cases regarding the thermal state of the surface were distinguished. In the following we consider the first two cases.
If the vehicle surface is radiation cooled and the heat flux into the wall, qw, is small, the radiation-adiabatic temperature Tw = Tra is to be determined. This is case 1:
qgw(x, Z) « aeT^ai’X, z) t Tw = Tra(x, Z) = ?
Case 2 is the case with Tw prescribed directly, because, for instance, of design considerations, or because the situation at a cold-wall wind-tunnel model is studied. Hence the heat flux in the gas at the wall, qgw, is to be determined. This is case 2:
Tw Tw (x: z) t qgw (x: z) ?
To describe the thermal state of the surface we remain with the reference temperature extension, following [16, 39], and perfect-gas flow. We begin with the discussion of the situation at flat surface portions, fp’. The situation at the stagnation-point region and at attachment lines is considered in SubSection 7.2.6.
The basis of the following relations is the Reynolds analogy, respectively the Chilton-Colburn analogy 
Summary. We summarize the dependencies in Table 7.7. We substitute also here ‘o’ conditions by general ‘ref ’ conditions and chose ш = =
0.65 in the viscosity law, Section 4.2. We break up the Reynolds numbers Re into the unit Reynolds number Reu and the running length x in order to show the explicit dependencies on these parameters. They reflect inversely the behavior of the boundary-layer thickness of laminar flow and the viscous sublayer thickness of turbulent flow, Sub-Section 7.2.1. As was to be expected, the qualitative results from Sub-Section 3.2.1 are supported. We see also that indeed in case 2 with given Tw the heat flux in the gas at the wall qgw has the same dependencies as T4a.
— Dependence on the boundary-layer running length x.
Both T4a, case 1, and qgw, case 2, decrease with increasing x. This is stronger in the laminar, (ж x—0 5), than in the turbulent cases (ж x—0 2).
Table 7.7. Flat surface portions: dependence of the thermal state of the surface, case 1 and case 2, on the boundary-layer running length x, the unit Reynolds number ReUef, the reference-temperature ratio T* /Tref, and the temperature difference Tr — Tw (w = = 0.65).
— Dependence on the unit Reynolds number ReUef.
The larger ReU, the larger T^a, case 1, and qgw, case 2, because the respective boundary-layer thicknesses become smaller with increasing ReЦ,. The thermal state of the surface with a laminar boundary layer reacts less strongly on changes of the unit Reynolds number, ж (Re%ef )0’5, than that of the turbulent boundary layer, ж (ReUef )0’8.
— Dependence on T*/Tref.
The larger T*/Tref, the smaller are T^a, case 1, and qgw, case 2. For a given Mref and a given Tref an increase of the wall temperature Tw would lead to a decrease of them. The effect is stronger for turbulent flow, ж (T*/Tref)-067, than for laminar flow, ж (T*/Tref)-0175. However, the major influence is that of the Tr — Tw, next item.
— Dependence on Tr — Tw.
at the attachment line of an infinite swept circular cylinder. We are aware that this is a more or less good approximation of the situation at stagnation points and (primary) attachment lines at a hypersonic flight vehicle in reality.
We find at the sphere, ‘sp’, respectively the circular cylinder (2-D case), for case 1 for perfect gas with the generalized reference-temperature formulation  like before
with C = 0.763 for the sphere and C = 0.57 for the circular cylinder. The velocity gradient due/dxx=0 is found with eq. (6.166). This is the laminar case.
The meanwhile classical formulation for qgw at the stagnation point of a sphere with given wall enthalpy hw, generalized case 2, in the presence of high-temperature real-gas effects is that of J. A. Fay and F. R. Riddell . This result of an exact similar solution ansatz reads
kPr-0’b(pw Pw )0A(PePe)0A
The stagnation point values are denoted here as boundary-layer edge ‘e’ values. For the sphere k = 0.763 and for the circular cylinder k = 0.57 . The term in square brackets contains the Lewis number Le, eq. (4.93). Its exponent is m = 0.52 for the equilibrium and m = 0.63 for the frozen case over a catalytic wall.
The term hD is the average atomic dissociation energy times the atomic mass fraction in the boundary-layer edge flow . For perfect-gas flow the value in the square brackets reduces to one. Note that in eq. (7.162) the external flow properties pepe have a stronger influence on qgw than the properties at the wall pw pw. This reflects the dependence of the boundary-layer thickness in the stagnation point on the boundary-layer edge parameters, eq. (7.133).
Because we wish to show the basic dependencies also here, we use the equivalent generalized reference-temperature formulation for case 2. The heat flux in the gas at the wall, qgw, for the sphere or the circular cylinder (2-D case) with given Tw, reads 
qgw, sp = kooPr1/3gsp—Tr(l – (7.163)
К T r
The radiation-adiabatic temperature, case 1, at the attachment line of an infinite swept circular cylinder, ‘scy’, is found from
where g*Scy is
4gw, scy = P)’3 kxgscy —Tr(l — – pjg – ). (7.166)
Like in Sub-Section 7.2.4, К is the radius of the cylinder, due/dxx=0 the gradient of the inviscid external velocity normal to the attachment line, eq. (6.167), and we = иж sin the inviscid external velocity along it, Fig. 6.37 b). For laminar flow C = 0.57, n = 0.5 and p* = p°’8pW2, = M°’8mW’2.
For turbulent flow C = 0.0345, n = 0.21. The reference-temperature values are found with eq. (7.151), and p* again with T* and the external pressure Pe.
Summary. In Table 7.8 the general dependencies of the radiation-adiabatic temperature, (case 1), are summarized for laminar and turbulent flow. Again we choose ш = = 0.65 in the viscosity law, Section 4.2. We break up,
like before, the Reynolds numbers Ke into the unit Reynolds number Keu and the radius К. The reference-temperature dependencies are also taken in simplified form. The dependencies of qgw (case 2) are the same, see eqs. (7.163) and (7.166) and are therefore not shown.
The results are:
— Dependence on the radius R.
The fourth power of the radiation adiabatic temperature T4a is the smaller, the larger К is. It depends (infinite swept circular cylinder) on К stronger for laminar, ж K—0 5, than for turbulent flow, ж K—0 21.
— Dependence on the sweep angle y>.
This dependence holds only for the infinite swept circular cylinder. For <p = 0° we have the case of the non-swept circular cylinder (2-D case). For <p ^ 90° T4a ^ 0. This means that we get the situation on an infinitely long cylinder aligned with the free-stream direction, where finally 5 and Svs ^ to, and hence T4a becomes zero.
Table 7.8. The radiation-adiabatic temperature, (case 1), at the sphere, respectively the circular cylinder (2-D case), and at the attachment line of the infinite swept circular cylinder: dependence on the radius R, the sweep angle y, the unit Reynolds number Re(^, and the reference-temperature ratio T* /Tx (ш = =
0.65). The temperature difference Tr – Tw = Tr — Tra is not included.
— Dependence on the unit Reynolds number ReU.
T4a depends on some power of the unit Reynolds number in the same way as on flat surface portions with ж (Re^,)0 5 for the laminar and ж (ReU)0’79 for the turbulent case. The larger ReU, the larger is T4a, because 6 and 6vs become smaller with increasing ReЦ,.
— Dependence on T*/Tr ef.
The larger T*/Tref, the smaller is T4a. For a given Mref and a given Tref an increase in wall temperature Tw would lead to a decrease of them. The effect is stronger for turbulent flow, ж (T*/Tref )-0’67, than for laminar flow, ж (T*/Tref )-0175. However, the major influence is that of the Tr — Tw, next item.
— Dependence on Tr — Tw (not in Table 7.8).