Answer:
Step-by-step explanation:
In a normal shock, a supersonic flow is compressed to a subsonic flow, resulting in an increase in static pressure and temperature of the gas. This change in momentum of the gas results in a force exerted on the surface of the duct.
To calculate the force exerted by the flow on the duct in the axial direction, we can use the conservation of mass, momentum, and energy across the normal shock. The pressure, temperature, and density ratios across the shock can be calculated using the Rankine-Hugoniot relations.
Assuming the upstream Mach number of the flow is known, we can calculate the downstream Mach number using the equation:
M2 = sqrt((1+((gamma-1)/2)M1^2)/(gammaM1^2-(gamma-1)/2))
where M1 is the upstream Mach number, M2 is the downstream Mach number, and gamma is the ratio of specific heats.
Using the downstream Mach number, we can calculate the pressure, temperature, and density ratios across the shock using the Rankine-Hugoniot relations:
P2/P1 = (2gammaM1^2 - (gamma-1))/(gamma+1)
T2/T1 = (2gammaM1^2 - (gamma-1))*(gamma-1)/(gamma+1)^2
rho2/rho1 = (gamma+1)*M1^2/((gamma-1)*M1^2+2)
where P1, T1, and rho1 are the upstream pressure, temperature, and density, respectively, and P2, T2, and rho2 are the downstream pressure, temperature, and density, respectively.
Once we have the pressure and density ratios, we can calculate the downstream velocity using the continuity equation:
A1V1 = A2V2
where A1 and A2 are the upstream and downstream cross-sectional areas, respectively.
Finally, we can calculate the force exerted by the flow on the duct in the axial direction using the momentum equation:
F = rho2A2(V1-V2)
where F is the force exerted on the duct, A2 is the downstream cross-sectional area, and V1 and V2 are the upstream and downstream velocities, respectively.
Note that the force exerted by the flow on the duct will be in the direction opposite to the flow direction, as the flow is decelerated in the normal shock.