Abstract
In a number of experimental and numerical publications a deviation has been found between the measured or computed stagnation point heat flux and that given by the theory of Fay and Riddell. Since the formula of Fay and Riddell is used in many applications to yield a reference heat flux for experiments performed in wind tunnels, for flight testing and numerical simulations, it is important that this reference heat flux is as accurate as possible. There are some shortcomings in experiments and numerical simulations which are responsible in some part for the deviations observed. But, as will be shown in the present paper, there is also a shortcoming on the theoretical side which plays a major role in the deviation between the theoretical and experimental/numerical stagnation point heat fluxes. This is caused by the method used so far to determine the tangential velocity gradient at the stagnation point. This value is important for the stagnation point heat flux, which so far has been determined by a simple Newtonian flow model. In the present paper a new expression for the tangential velocity gradient is derived, which is based on a more realistic flow model. An integral method is used to solve the conservation equations and, for the stagnation point, yields an explicit solution of the tangential velocity gradient. The solution achieved is also valid for high temperature flows with real gas effects. A comparison of numerical and experimental results shows good agreement with the stagnation point heat flux according to the theory of Fay and Riddell, if the tangential velocity gradient is determined by the new theory presented in this paper.
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Abbreviations
- D:
-
sphere diameter
- F:
-
variable
- H:
-
reduced enthalpy=h/RT ref , R=288 Nm/kgK, T ref =273.15 K
- Le:
-
Lewis number
- p:
-
pressure
- P u :
-
\(\bar \rho \bar u^2 + 2p_s \bar p/(\rho _s u_{max}^2 )\)
- P uv :
-
\(\bar \rho \bar u\bar v\)
- P v :
-
\(\bar \rho \bar v^2 + p_s \bar p/(\rho _s u_{max}^2 )\)
- Q u :
-
\(\bar \rho \bar u\)
- Q v :
-
\(\bar \rho \bar v\)
- R:
-
sphere radius
- r,Θ :
-
polar coordinates
- u :
-
velocity component in circumferential direction
- u ∞ :
-
free stream velocity
- v :
-
radial velocity component
- Δ :
-
shock stand-off distance
- ε :
-
ρ ∞/ρ2, density ratio across the shock
- η :
-
r·sinΘ, coordinate normal to the sphere axis
- μ :
-
viscosity
- ρ :
-
density
- σ :
-
local shock angle
- Ω :
-
reaction rate
- b:
-
body
- max:
-
maximum adiabatic velocity
- s:
-
boundary layer edge, stagnation point
- w:
-
wall
- o:
-
reservoir condition
- 2:
-
conditions immediately behind the shock
- ∞:
-
free stream quantity
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Olivier, H. Influence of the velocity gradient on the stagnation point heating in hypersonic flow. Shock Waves 5, 205–216 (1995). https://doi.org/10.1007/BF01419002
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DOI: https://doi.org/10.1007/BF01419002