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MECH5304 - Assignment 2 (Due March 19, 2012, max. Analytical Solution Consider the steady, isentropic ow through a convergent-divergent nozzle. The...

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MECH5304 – Assignment 2 (Due March 19, 2012, max. 4 pages) For this assignment, you will use MacCormack’s second-order two-step finite-difference scheme to solve the convergent-divergent nozzle problem consisting of the one-dimensional continuity, momentum, and energy equations. Analytical Solution Consider the steady, isentropic flow through a convergent-divergent nozzle. The gov- erning equations are: Continuity ρ 1 V 1 A 1 = ρ 2 V 2 A 2 (1) Momentum p 1 A 1 + ρ 1 V 2 1 A 1 + A 1 Z A 2 pdA = p 2 A 2 + ρ 2 V 2 2 A 2 (2) Energy h 1 + V 2 1 2 = h 2 + V 2 2 2 (3) where subscripts 1 and 2 denote different locations along the nozzle. The perfect gas equation of state is p = γRT (4) and for a calorically perfect gas, we have h = c P T (5) Equations (1) to (5) can be solved analytically ± A A * ² 2 = 1 M 2 ³ 2 γ + 1 ± 1 + γ - 1 2 M 2 ²´ ( γ +1) / ( γ - 1) (6) p p 0 = ± 1 + γ - 1 2 M 2 ² - γ/ ( γ - 1) (7) ρ ρ 0 = ± 1 + γ - 1 2 M 2 ² - 1 / ( γ - 1) (8) T T 0 = ± 1 + γ - 1 2 M 2 ² - 1 (9) where γ = c p /c v = 1 . 4 for air at standard conditions.
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2 Numerical Solution We have Continuity ∂ρA ∂t + ρA ∂V ∂x + ρV ∂A ∂x + V A ∂ρ ∂x = 0 (10) Momentum ρ ∂V ∂t + ρV ∂V ∂x = - R ± ρ ∂T ∂x + T ∂ρ ∂x ² (11) Energy ρc ν ∂T ∂t + ρV c ν ∂T ∂x = - ρRT ± ∂V ∂x + V (ln A ) ∂x ² (12) Defining non-dimensional variables: T 0 = T/T 0 0 = ρ/ρ 0 ,x 0 = x/L,V 0 = V/a 0 ,t 0 = t/ ( L/a 0 ) ,A 0 = A/A * for temperature, density, axial length, velocity, time and area, respectively, where T 0 , ρ 0 and a 0 = γRT 0 are the temperature, density and speed of sound in the reservoir, L is a reference length for the nozzle and A * is the sonic throat area. Then Continuity ∂ρ 0 ∂t 0 = - ρ 0 ∂V 0 ∂x 0 - ρ 0 V 0 (ln A 0 ) ∂x 0 - V 0 ∂ρ 0 ∂x 0 (13) Momentum ∂V 0 ∂t 0 = - V 0 ∂V 0 ∂x 0 - 1 γ ± ∂T 0 ∂x 0 + T 0 ρ 0 ∂ρ 0 ∂x 0 ² (14) Energy ∂T 0 ∂t 0 = - V 0 ∂T 0 ∂x 0 - ( γ - 1) T 0 ± ∂V 0 ∂x 0 + V 0 (ln A 0 ) ∂x 0 ² (15) Finite Difference Equations Dropping the prime (meaning dimensionless variable) for the sake of simplicity and using MacCormack’s predictor-corrector technique, we have Continuity ± ∂ρ ∂t ² t i = - ρ t i V t i +1 - V t i Δ x - ρ t i V t i ln A i +1 - ln A i Δ x - V t i ρ t i +1 - ρ t i Δ x (16) Momentum ± ∂V ∂t ² t i = - V t i V t i +1 - V t i Δ x - 1 γ ± T t i +1 - T t i Δ x + T t i ρ t i ρ t i +1 - ρ t i Δ x ² (17) Energy ± ∂T ∂t ² t i = - V t i T t i +1 - T t i Δ x - ( γ - 1) T t i ± V t i +1 - V t i Δ x + V t i ln A i +1 - ln A i Δ x ² (18)
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