Commercial Space Exploration: Risks, Benefits, and Mars Base Design
The commercial exploration of space presents various risks, such as space piracy and extreme conditions, along with benefits like mining opportunities and technological advances. Design competitions for Mars bases aim to support human exploration on the planet, considering Mars' unique atmosphere and conditions. Priorities for a Mars base include addressing essential needs, mitigating challenges through careful design, and navigating compromises for the best design. Explore the possibilities and challenges of commercial space endeavors.
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Presentation Transcript
Stream Function & Velocity Potential Stream lines/ Stream Function ( ) Concept Relevant Formulas Examples Rotation, vorticity Velocity Potential( ) Concept Relevant Formulas Examples Relationship between stream function and velocity potential Complex velocity potential
Stream Lines Consider 2D incompressible flow Continuity Eqn ( ) ( ) ( ) + + + = 0 V V V x y z t x y z V x ( ) V ( ) = + = x V dy 0 V y x y x y Vx and Vy are related Can you write a common function for both?
Stream Function Assume = x V y Then 2 V x = = x V dy dy y x y 2 = = dy y x x Instead of two functions, Vx and Vy, we need to solve for only one function Stream Function Order of differential eqn increased by one
Stream Function What does Stream Function mean? Equation for streamlines in 2D are given by = constant Streamlines may exist in 3D also, but stream function does not Why? (When we work with velocity potential, we may get a perspective) In 3D, streamlines follow the equation dx V dy V dz V = = x y z
Rotation Definition of rotation V xy + y Time=t y y V x xy V V yx yx + x x Assume Vy|x < Vy|x+ x + d dt = = ROTATION z 2 and Vx|y > Vx|y+ y
Rotation To Calculate Rotation y x = tan 1 ( ) ( ) y1 = y V t V t 1 y y + x x x x ( ) + V V t y y = x x x arctan x ( ) Similarly V V t x x + y y y = arctan y
Rotation To Calculate Rotation ( ) ( ) + + + d dt 1lim 2 = = = + t t t ROTATION z t 2 0 t ( ) ( ) V V t V V t y y x x + y y y + x x x arctan arctan x y 1 2 1 2 = lim t x y lim t t t 0 0 0 0 0 0 x y For very small time and very small element, x, y and t are close to zero
Rotation To Calculate Rotation For very small (i.e. ) sin tan cos 1 arctan ( ) ( ) V V t V V t y y y y + + x x x x x x arctan x x ( ) ( ) V V t V V t y y y y + + x x x x x x arctan x x = lim t x y lim t x y t t 0 0 0 0 0 0
Rotation To Calculate Rotation ( lim x ) V V V x y y + y = x x x x ( 0 ) ( ) V V t V V t y y x x + y y y + x x x arctan arctan x y 1 2 1 2 = lim t x y lim t x y z t t 0 0 0 0 0 0 Simplifies to 1 2 ( ) V x 1 2 V y = V y = x z z
Rotation in terms of Stream Function To write rotation in terms of stream functions, Assume that is a stream function. So = = x V V y y x V x 2 2 1 2 1 2 V y 1 2 y = = x z 2 2 x y = 2 That is For irrotational flow ( z=0) 2 + = 2 0 z 2 = 0
Stream Function- Physical meaning Statement: In 2D (viscous or inviscid) flow (incompressible flow OR steady state compressible flow), = constant represents the streamline. Proof If = constant, then d = ( 0 ( ) V dy = + d dx dy x y ) = + V dx y x = Vy If = constant, then V V Vx dy dx y = x
