Stagnation Point Aerodynamic Heating
Contrary to what you might think, even though the flow velocity is zero at a stagnation point, the boundary layer at the stagnation point can be defined and has a finite thickness. The flow conditions at the edge of the stagnation point boundary layer are given by the inviscid solution for a stagnation point; in particular, at the boundary-layer
edge, the velocity is zero and the temperature is the total temperature, that is, ue = 0 and Te = To. This is shown in Figure 18.10. Moreover, along the vertical line in the ^-direction shown in Figure 18.10, и = 0 at every point inside the boundary layer. However, the ratio (и/и,,) — (0/0) is an indeterminant form that has a finite value at each point in the boundary layer. As in the case of the flat plate solutions discussed in Sections 18.2 and 18.3, we define a function /(r?) such that (u/ue) = and
/’ has a definite profile through the boundary layer. Indeed, we can define the edge of the boundary layer as the point where (u/ue) = /'(fl) = 0.99. Finally, we note that the shear stress at the wall at the stagnation point (point A in Figure 18.10) is zero. This not only comes out of the solution of the boundary layer equations, but it is obvious by inspection. Along the wall above point A the shear stress acts upward, and below point A it acts downward. Hence right at point A the shear stress must go through zero.
If the above discussion sounds rather theoretical, the temperature profile through the stagnation point boundary layer is easier to visualize. The temperature at the outer edge is the total temperature 7b. The temperature at the wall at r] — 0 is Tw. Hence, there is a temperature profile that exists in the normal direction through the stagnation point boundary layer. The heat transfer at the stagnation point is given by
the temperature gradient at point A, namely,
The practical purpose of a stagnation point boundary-layer solution is to calculate the heat transfer, qw.
The boundary-layer equations, Equation (17.28)—(17.31) applied at the stagnation point region are transformed using a version of the transformation described in Section 18.3, namely,
where h is the static enthalpy (since и = 0, the static and total enthalpies are the same). This leads to the stagnation point boundary layer equations given below. For a detailed derivation of these equations, see for example Chapter 6 of Reference 55.
where C = (pfi/pefie). Equations (18.63) and (18.64) are the governing equations for a compressible, stagnation-point boundary layer. Examining these equations, we see no § -dependency. Hence, the stagnation point boundary layer is a self-similar case.
Numerical solutions to Equations (18.63) and (18.64) can be obtained by the “shooting technique” as described earlier in the flat plate case. There is nothing to be gained in going through the details at this stage of our discussion. Instead, we simply state the result of solving Equations (18.63) and (18.64), correlated in the following expression obtained from Reference 82:
Cylinder: qw = 0.57 Рґ0’6(редг)І/2./^(/і„ – hw) [18.65]
If we had considered an axisymmetric body, the original transformation given by Equations (18.59) and (18.60) would have been slightly modified as follows:
1 = peueper2 dx Jo
where r is the vertical coordinate measured from the centerline, as shown in Figure 18.10. Equations (18.66) and (18.67) lead to equations for the axisymmetric stagnation point almost identical to Equation (18.63) and (18.64), namely,
[18.69] where C = (p/і/pe/ie). In turn, the resulting heat transfer expression is (Reference 82):
Compare Equation (18.65) for the two-dimensional cylinder with Equation
(18.70) for the axisymmetric sphere. The equations are the same except for the leading coefficient, which is higher for the sphere. Everything else being the same, this demonstrates that stagnation point heating to a sphere is larger than to a twodimensional cylinder. Why? The answer lies in a basic difference between two – and three-dimensional flows. In a two-dimensional flow, the gas has only two directions to move when it encounters a body—up or down. In contrast, in an axisymmetric flow, the gas has three directions to move—up, down, and sideways—and hence the flow is somewhat “relieved,” that is, in comparing two – and three-dimensional flows over bodies with the same longitudinal section (such as a cylinder and a sphere), there is a well-known three-dimensional relieving effect for the three-dimensional flow. As a consequence of this relieving effect, the boundary-layer thickness 8 at the stagnation point is smaller for the sphere than for the cylinder. In turn, the temperature gradient at the wall, (dT/dy)w, which is of the order of (Te/8), is larger for the sphere. Since qw = k(dT/dy)w, then qw is larger for the sphere. This confirms the comparison between Equations (18.65) and (18.70).
The above results for aerodynamic heating to a stagnation point have a stunning impact on hypersonic vehicle design, namely, they impose the requirement for the vehicle to have a blunt, rather than a sharp, nose. To see this, consider the velocity gradient, due/dx, which appears in Equations (18.65) and (18.70). From Euler’s equation applied at the edge of the boundary layer
Assuming a newtonian pressure distribution over the surface, we have from Equation
Cp = 2 sin2 в
where 9 is defined as the angle between a tangent to the surface and the freestream direction. If we define ф as the angle between the normal to the surface and the freestream, then Equation (14.4) can be written as
Cp =2 cos2 ф
From the definition of Cp, Equation (18.73) becomes
Differentiating Equation (18.74), we obtain
dPe Л Л ■ л^
‘ = —4<7oc cos ф sin 0 —
Combining Equations (18.72) and (18.75), we have
Equation (18.76) is a general result which applies at all points along the body. Now consider the stagnation-point region, as sketched in Figure 18.11. In this region, let Ax be a small increment of surface distance above the stagnation point, corresponding to the small change in ф, Аф. The inviscid velocity variation in the stagnation region can be shown to be
Also, in the stagnation region ф is small, hence, from Figure 18.11,
cos ф ~ 1
sin ф ~ ф % Аф ~ —–
where R is the local radius of curvature of the body at the stagnation point. Finally, at the stagnation point, Equation (18.73) becomes
Substituting Equations (18.77)—(18.81) into (18.76), we have
/duA2 _ 2(pe – poo) /Ax / J dx ) pe Ay R ) R)
Examine Equations (18.65) and (18.70) in light of Equation (18.82). We see that
This states that stagnation-point heating varies inversely with the square root of the nose radius; hence, to reduce the heating, increase the nose radius. This is the reason why the nose and leading edge regions of hypersonic vehicles are blunt; otherwise, the severe aerothermal conditions in the stagnation region would quickly melt a sharp leading edge.
Return to Section 1.1 and review our qualitative discussion contrasting the aerodynamic heating for slender and blunt reentry vehicles. There we argued on a qualitative basis that to minimize aerodynamic heating a blunt nose must be used. We have now quantitatively proven this fact with the derivation of Equation (18.83).
The fact that qw is inversely proportional to ~/R is experimentally verified in Figure 18.12, obtained from Reference 83. Here, various sets of experimental data for Ся at the stagnation point are plotted versus Reynolds number based on nose diameter; the abscissa is essentially proportional R. This is a log-log plot, and the data exhibits a slope of —0.5, hence verifying that qw ос 1 //R.
Figure 18.13 Stagnation point Stanton number versus Re based on nose radius. (Source: Koppenwallner, Reference 83.)