Stream Function in Polar Coordinates
Learn about stream functions in polar coordinates, source and sink flow fields, useful combined flow field scenarios, and the equation for rectilinear flow. Explore the relationship between source and sink strengths, circular streamlines, and more.
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Presentation Transcript
Stream function in polar coordinate ??? ????????? ?????????? ? = ?(?,?) With ? = ??=?? ?? ??? ? = ??= ?? ?? ???? +?? ???? + ?? ??? polar coordinate ? = ?(?,?) d? =?? ???y = -v dx + u dy ? = ?? ???y + C = ? ??+ u dy + ? ?? ??? ??? ??= ?? ?????? +?? With ??= ?? ?? ????= ????? - ??dr d? = ?? ?????? + ?? ???? + C= ????? - ?? dr + C ? =
Basic Flowfields ?= ? ?? Source 2- Source and Sink 2-1 Source For the following flowfield ??= ?? ???= ? ?= ? ?= ? 2 ? ??? ??= 0 ? ?? ?????? + ?? ???? + C ? = = ????? - ?? dr + C = ( = ? Let ?=0 at ? = 0 0 = ? C = 0 which means that ?= ? ? 2 ?)??? - (0) dr + C 2 ? + C ?= ?? ? ?= -? 2 0 + C ?? Sink ??? 2-2 Sink In this case the stream function will be the inverse of source function ?=-? ?= ? ? ?= ?? ? ?= - ??? ?
Some Useful Combined Flowfields 2- Source and Sink of Equal Strength F?? ?????? ??= ? For sink ??= - ???? ???? ? For combined flow For combined flow ? = ??+ ?? or ? = ? = ? For any point P , note that for the angle ( For any point P , note that for the angle (?) between r r1 1 and r and r2 2 is is ? = ?? ?? ? ???? ?? ( ?? ??) ???? = between ? ? = - For For constant constant ? at any stream line, at any stream line, the constant . That means all the stream line is circles constant . That means all the stream line is circles through the source and sink through the source and sink ??? the ? ? is also a is also a ??? ????????? ?????????? ??= ??? ? ? ? ?+? and ??= ??? ? ? ? ? ? ? ? = ????? ? ?+? ??? ? ? ?
Some Useful Combined Flowfields 3- Source and Sink of Equal Strength in Rectilinear Flow F?? Source and Sink of Equal Strength ??= ? For Rectilinear Flow??= U y ? ? ????? ? ?+? ??? ? ? ? So, For combined flow So, For combined flow ? = ??+ ?? ? ? ? or ? = ????? ? ?+? ??? ? ? ?+? ? ?+? ) the velocity u = 0 0 ? (? ? ?) ? ??? ( prove it ) ( prove it ) L/2 2 , , 0 0) ) ? = + U y + U y ? ? ? = ? ? = ??=?? ? ? ? ?? = ? + + U U ?? ? ?+ ?+ ? ? At point At point s sL L ( (- -L/ 0 = ?? L/2 2 , , 0 0) the velocity u = ? ?+?) = ? + + U U (? Or ? ?= ? ? + Or ? ) = ? ) = ? ????? ?? ??? ?? at this at this point point s sL L ( (- -L/ + U + U ( (0 0) = ?? ( (?) = ?
Some Useful Combined Flowfields For the equation of the body For the equation of the body ? ?= ? ????? ? The distance b could be find from the point ( The distance b could be find from the point (0 0, b/ ? = ? ? ? ????? ? ?+? ??? ? ? ?+? ??? ? + U y + U y ? ? ? ? Or Or 0 0 = = + U + U y y - - ? ? ?? , b/2 2) as follows ) as follows ? ? ? + U + U y + U ? ? ? y - -? ????? ? ?+? ??? ? From From ? ? ?/? ?+? ??? ? ? ?? ??? ? ? ??+ ??? ? ??? ??? ? ?/? ? ? ? ?? ? ?? + U ? + U ? ? - -? ? - -? ? - -? ? = ????? ? ? ????? ? ? ????? ? ? = ? ???? ? + U ? + U + U ? ? = ? + U + U ? ? = ? ? ? ?? ? ??+ U ? - - ? ? - - ? + U ? ? = ? Solve by trail and error to find b/ Solve by trail and error to find b/2 2 Note that Note that ? = ?? ???? ??(?? ?? ?? ??)
