Consider the supersonic flow over a flat plate at an angle of attack. Outline a method to calculate the strengths of trailing edge shock and expansion waves, and direction of the flow downstream of the trailing edge.
Outline a method to calculate the strengths of trailing edge shock and expansion waves , and direction of the flow downstream of the trailing edge.
The supersonic flow over a flat plate at an angle of attack in a free stream with given Mach number, M∞ is sketched below
The flow direction downstream of the leading edge is given by line ab. The flow direction is below the horizontal (below the direction of M∞) because lift is produced on the flat plate, and due to overall momentum considerations, the downstream flow will be inclined slightly downward.
Line ab is a slip line. Since flows over top and bottom of plate have gone through shock waves of different strengths entropy in region 4 and 5 are different.
The boundary conditions that must hold across the slip line is constant pressure that P4 will be equal to P5 and strengths of trailing edge shock waves and expansion waves is fixed by this boundary condition.
We can use iterative approach in order to calculate flow direction and strength of trailing edge shocks and expansion waves.
- a) We have to assume a value of ∅.
- b) In order to obtain P4, we need to calculate strength of trailing edge shock by using local deflection angle (α – ∅).
- c) In order to obtain P5, we need to calculate strength of trailing edge expansion wave for a local expansion angle (α – ∅).
- d) If P4 and P5 are different than we have to assume a new value of ‘∅’.
- e) When P4 = P5 , than condition is satisfied and iteration is said to be converged, trailing edge flow is determined.
