Orthogonal Polynomial Solution of the Boltzmann Equation for a Strong Shock Wave

1966 ◽  
Vol 9 (9) ◽  
pp. 1621 ◽  
Author(s):  
Shelden H. Radin
Keyword(s):  
Shock Wave ◽  
Boltzmann Equation ◽  
Orthogonal Polynomial ◽  
Polynomial Solution ◽  
Strong Shock ◽  
Strong Shock Wave ◽  
The Boltzmann Equation

Solution of the Boltzmann equation at the upstream and downstream singular points in a shock wave

1974 ◽  
Vol 65 (3) ◽  
pp. 603-624 ◽  
Author(s):  
J. P. Elliott ◽  
D. Baganoff
Keyword(s):  
Shock Wave ◽  
Boltzmann Equation ◽  
Mach Number ◽  
Singular Point ◽  
Shock Waves ◽  
Strong Shock ◽  
Singular Points ◽  
The Boltzmann Equation ◽  
Closure Relations ◽  
Strong Shock Waves

A solution of the Boltzmann equation is obtained at the upstream and downstream singular points in a shock wave, for the case of Maxwell molecules. The fluid velocityu, rather than the spatial co-ordinatex, is used as the independent variable, and an equation for ∂f/∂uat a singular point is obtained from the Boltzmann equation by taking the appropriate limit. This equation is solved by using the methods of Grad and of Wang Chang & Uhlenbeck; and it is observed that the two methods are the same, since they involve not only an equivalent system of moment equations but also the same closure relations. Because many quantities are zero at a singular point, the problem becomes sufficiently simple to allow the solution to be carried out to any desired order. At the supersonic singular point, the solution converges very slowly for strong shock waves; but a simple modification to Grad's method provides a rapidly convergent solution. The solution shows that the Navier-Stokes relations, or the first-order Chapman-Enskog results, do not apply unless the shock-wave Mach number is unity, and that they are grossly in error for strong shock waves. The solution confirms the existence of temperature overshoot in a strong shock wave; shows that the critical Mach number in Grad's solution increases monotonically with the order of the solution; provides a simple explanation as to why Grad's closure relations fail and shows how they can be improved; and provides exact boundary values that can be used to guide future numerical solutions of the Boltzmann equation for shock-wave structure.

Sporadic interaction of a strong shock wave with a thin cone

1977 ◽  
Vol 11 (6) ◽  
pp. 919-924
Author(s):  
V. I. Bogatko ◽  
G. A. Kolton
Keyword(s):  
Shock Wave ◽  
Strong Shock ◽  
Strong Shock Wave

Axisymmetric hypersonic flow with strong viscous interaction

1968 ◽  
Vol 34 (4) ◽  
pp. 687-703 ◽  
Author(s):  
John Webster Ellinwood ◽  
Harold Mirels
Keyword(s):  
Shock Wave ◽  
Hypersonic Flow ◽  
Power Law ◽  
Axisymmetric Flow ◽  
Strong Shock ◽  
The Body ◽  
Strong Shock Wave ◽  
Viscous Interaction ◽  
Slender Bodies ◽  
Prandtl Numbers

Stewartson's theory for axisymmetric hypersonic flow of a model gas over slender bodies with strong viscous interaction and strong shock wave is extended to power-law viscosity variation and Prandtl numbers other than one. Flow properties at the body surface and shock are obtained without recourse to numerical integration. Numerical computations are presented for axisymmetric flow over a three-quarter power-law body with strong shock wave and viscous interactions that range from weak to strong.

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