AERO96001 - Method of Characteristics for Steady 2D Supersonic Flow
Equation which defines characteristic lines
(dy/dx)|characteristic = tan(θ±µ) (See page 96-97 for derivation, also page 97 for graphical representation)
Notation of characteristic lines in supersonic flow and why
+µ and -µ characteristic lines Since these characteristic lines are always inclined at an angle of ±µ to the flow +µ characteristic lines are the same as left-running waves, -µ are the same as right-running waves
Region of dependence and influence
Draw +µ and -µ characteristics through point Domain of dependence is quadrant upstream of point Domain of influence is quadrant downstream of point (See page 99 for diagram)
Steps for MoC for steady 2D supersonic flow
Find characteristic lines in a flow where properties are continuous but derivatives are indeterminate: tan⁻¹(dy/dx) = θ±µ Determine compatibility equations that hold along these characteristic lines: ν±θ = constant Solve equations along these lines step by step, starting from known initial conditions
Key assumption for application of MoC
Flow must be irrotational - from Crocco's relation, can prove that steady, irrotational flow with a uniform freestream is isentropic - entropy is constant in the region of interest
Where can constant pressure boundaries occur?
Isentropic transonic flows - the boundary between supersonic and subsonic flow, known as the sonic line, is a constant pressure boundary
Does Mach number either side of sonic line have to be equal?
No - e.g. in over and under-expanded jets, stagnation pressure of exiting jet is always greater than that of surrounding fluid In this case, a constant pressure boundary still exists to separate the supersonic region from the surrounding stationary air (See page 102 for diagrams)
Final result of compatibility equation for MoC for 2D steady flows
Previous expression is identical to Prandtl-Meyer expression - can be integrated along characteristic lines to give: Along -µ characteristics: θ+ν = constant = K_ Along +µ characteristics: θ-ν = constant = K₊ K₊ and K_ are analogous to Riemann invariants J_ and J₊ from 1D unsteady flow
Process to apply MoC to 2D steady flows
Same procedure as 1D unsteady flows: Start from initial location where conditions are known and move along characteristic lines to determine conditions at subsequent points in the flow
Two types of boundary conditions in 2D steady supersonic flow
Solid walls Constant pressure boundaries
Crocco's Theorem and how it shows that steady, irrotational flow will be isentropic
T.∇s + \bar{V} x (∇x\bar{V}) = ∇h₀ + ∂\bar{V}/∂t ∇h₀ = 0 because adiabatic ∂\bar{V}/∂t = 0 because steady ∇x\bar{V} = 0 because irrotational ∴ ∇s must = 0 and ∇s is entropy gradient ∴ isentropic
How are characteristics reflected when they are incident on a solid wall
c₊ (+µ) characteristics reflect as c_ (-µ) c_ (-µ) characteristics reflect as c₊ (+µ characteristics) Compressive characteristics reflect as compressive characteristics Expansive characteristics reflect as expansive characteristics
How are characteristics reflected when they are incident on a constant pressure boundary
c₊ (+µ) characteristics reflect as c_ (-µ) characteristics c_ (-µ) characteristics reflect as c₊ (+µ) characteristics Compressive characteristics reflect as expansive characteristics Expansive characteristics reflect as compressive characteristics
Compatibility equation for variation in flow properties along characteristic lines
dθ = -√(M²-1) dv/v when (dy/dx)|char = tan(θ-µ) dθ = √(M²-1) dv/v when (dy/dx)|char = tan(θ+µ) (See page 98 for derivation)
